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This is to certify that the
dissertation entitled
A Gauss-Galerkin Finite Element Method
for a Class of Singular Diffusion Equations
in Two Space Variables

presented by
Li Liu

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Applied Mathematics

 

 

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[hue December 301 1997

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A GAUSS-GALERKIN FINITE ELEMENT
METHOD FOR A CLASS OF SINGULAR
DIFFUSION EQUATIONS IN TWO SPACE
VARIABLES

Li Liu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1998

ABSTRACT

A GAUSS-GALERKIN FINITE ELEMENT
METHOD FOR A CLASS OF SIN GULAR
DIFFUSION EQUATIONS IN TWO SPACE
VARIABLES

By

M Liu

In this dissertation we are concerned with a Gauss-Galerkin finite element method
for a class of singular diffusion equations in two space variables. More specifically, we

consider the Fokker-Planck equation

Bu _ 6(au) +162(b2u) 6(cu) +162(d2u)

5t— _ 6x 2 63:2 _ 6y 2 ayz (3:,y,t) 6 (0’ 1)? x [OaTl-

 

 

We are concerned with the case when the problem is “singular” in the x variable and
“regular” in the y variable, i.e., the coefficient b2 may vanish along a: = 0 and a: = 1,
but c and d2 are bounded away from zero for y 6 [0,1].

In the proposed Gauss-Galerkin finite element method, finite element discretization
is made in the y variable and the Gauss-Galerkin method is used in the x variable be-
cause of the nature of the problem. Convergence of the finite element approximation

is established first. Then we analyze the convergence of the resulting Gauss-Galerkin

approximation. Finally, by combining the above results, the convergence of the Gauss-
Galerkin finite element approximation is obtained.

A number of test problems are studied. The numerical results show that the pro-
posed Gauss-Galerkin finite element method is efficient in solving singular diffusion

equations of the type considered here.

To my parents, my wife and my son

iv

ACKNOWLEDGMENTS

I would like to extend my gratitude and thanks to my advisor, Professor David
H.Y. Yen, for his suggestion of the dissertation problem and the helpful directions he
has proposed in solving it. His advice and encouragement are greatly appreciated. I
would also like to extend my warm thanks to other members of my guidance commit-
tee: Professors Tien-Yien Li, Gerald D. Ludden, Habib Salehi and Zhengfang Zhou
for their advice and helpful suggestions. Most of all, I want to acknowledge and thank
the people whose lives were most affected by this work: my wife, Bei Zhang, to whom
I offer the greatest thanks for her taking on all household and child-minding duties
as I spent long hours at work, my son, Terrance Liu, for his ability to make me focus
on what was important, and my parents, Xianrong Liu and Changqun Song, for their
unconditional pride and belief in me. Thank you, I couldn’t have done it without
your overwhelming support, encouragement, understanding and patience.

TABLE OF CONTENTS

LIST OF TABLES viii
LIST OF FIGURES xi
1 Introduction 1
1.1 Background and Motivation ....................... 1
1.2 Organization of the Dissertation ..................... 5
2 Preliminaries 7
2.1 Markov Process .............................. 7
2.2 Transition Probability .......................... 8
2.3 Diffusion Process ............................. 9
2.4 Fokker-Planck Equation for a Diffusion Process ............ 10
2.5 Stochastic Differential Equation ..................... 12
2.6 Existence and Uniqueness ........................ 13
3 Problem Formulation and Mathematical Properties 14
3.1 Initial-Boundary Value Problems and Their Weak Formulation . . . . 14
3.2 Mathematical Analysis of Initial-Boundary Value Problem ...... 19
4 Gauss-Galerkin Finite Element Method 27
4.1 Finite Element Approximation in the y Variable ............ 27
4.2 Gauss-Galerkin Approximation in the x Variable ............ 34
5 Convergence Results 51
5.1 Convergence of Semi-Discrete Finite Element Approximation in y . . 51
5.2 Convergence of the Gauss-Galerkin Approximations in x ....... 57
6 Numerical Results 73
6.1 Piecewise Linear Finite Element Space ................. 73
6.2 A Test Problem .............................. 81
6.3 Dependence of Solution upon Parameters in the Singularities ..... 93

vi

6.4 Dependence of Solution upon Lower Order Terms ........... 123
7 Conclusions and Discussions 149

BIBLIOGRAPHY 151

vii

6.1

6.2

6.3

6.4

6.8

6.9

6.10

6.11

6.12

LIST OF TABLES

Gauss-Galerkin Finite Element Method: Changes of the three nodes
#, + and x at y = 0.5 as t increases ...................
Gauss-Galerkin Finite Element Method: Changes of the weights at the
three nodes at, + and x at y = 0.5 as t increases ............
Gauss-Galerkin Finite Element Method: Changes of the “moment”
m§,(y, t) with three nodes *, + and x at y = 0.5 as t increases

GaussGalerkin Finite Element Method: Changes of the “total mo-
ment" 11;“) with three nodes at, + and x at as t. increases ......
Gauss-Galerkin Finite Element Method: Changes of the five nodes 0,
o, x, + and t at y = 0.5 as t increases ..................
Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes 0, o, x, + and at at y = 0.5 as t increases ............
Gauss-Galerkin Finite Element Method: Changes of the “moment”
mf,(y, t) with five nodes 0, o, x, + and a: at y = 0.5 as t increases . .
Gauss-Galerkin Finite Element Method: Changes of the “total mo-
ment” M},(t) with five nodes 0, o, x, + and :- as t increases .....
Gauss-Galerkin Finite Element Method: Changes of the exact total
moment M i(t) as t increases ......................
Gauss-Galerkin Finite Element Method: Changes of the errors between
the exact total moment M‘It) and their approximation Mn(t) as t
increases .................................
Gauss-Galerkin Finite Element Method: Changes of the relative errors
between the exact total moment M ‘(t) and their approximation M..(t)
as t increases ...............................
Gauss—Galerkin Finite Element Method: Changes of the nodes at, +
andxaty=0.5withp=2andq=2astincreases .........

viii

91

92

94

95

96

97

98

99

100

101

6.13 Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes't,+andxaty=0.5withp=2andq=2astincreases .. .
6.14 Gauss-Galerkin Finite Element Method: Changes of the “moment”
m§,(y,t) at y=0.5 withp=2 and q: 2 as t increases ........
6.15 Gauss-Galerkin Finite Element Method: Changes of the “total mo-
ment” Mf,(t) with p = 2 and q = 2 as t increases ...........
6.16 Gauss-Galerkin Finite Element Method: Changes of the nodes *, +
andxaty=0.5 withp=1andq=2astincreases .........
6.17 Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes #, +andxaty==0.5withp= 1 andq=2astincreases
6.18 Gauss-Galerkin Finite Element Method: Changes of the “moment”
mj,(y, t) at y = 0.5 with p = 1 and q = 2 as t increases ........
6.19 Gauss-Galerkin Finite Element Method: Changes of the “total mo-

ment” Mf,(t) at y = 0.5 with p = l and q = 2 as t increases ..... q

6.20 Gauss-Galerkin Finite Element Method: Changes of the nodes at, +
and x at y = 0.5 with p = 2 and q = 1 as t increases .........
6.21 Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes at, + and x at y = 0.5 with p = 2 and q = l as t increases ...
6.22 Gauss-Galerkin Finite Element Method: Changes of the “moment”
mf,(y, t) at y = 0.5 with p = 2 and q = 1 as t increases ........
6.23 Gauss-Galerkin Finite Element Method: Changes of the “total mo-
ment” Mf,(t) at y = 0.5 with p = 2 and q = 1 as t increases .....
6.24 Gauss-Galerkin Finite Element Method: Changes of the nodes t, +
and x at y = 0.5 with p = land q = 1.5 as t increases ........
6.25 Gauss-Galerkin Finite Element Method: Changes of the nodes at, +
andxaty=0.5 withp=landq=2.5astincreases ........
6.26 Gauss-Galerkin Finite Element Method: Changes of the nodes 1!, +
andxaty=0.5 withp=landq=3astincreases .........
6.27 Gauss-Galerkin Finite Element Method: Changes of the nodes at, +
andxaty=0.5 withp=landq=4astincreases .........
6.28 Gauss-Galerkin Finite Element Method: Changes of the nodes at, +
andxaty=0.5withp=1andq=lastincreases .........
6.29 Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes t, + andxaty=0.5withp= l andq= 1 astincreases
6.30 Gauss-Galerkin Finite Element Method: Changes of the “moment”
m§,(y,t) at y=0.5 withp=l and Q: l as t increases ........

108

109

116

117

129

130

6.31

6.32

6.33

6.34

6.35

6.36

6.37

6.38

6.39

Gauss-Galerkin Finite Element Method: Changes of the “total mo-
ment”‘Mf,(t) aty=0.5 withp=1andq=1astincreases .....
Gauss-Galerkin Finite Element Method: Changes of the nodes at, +
and x at y = 0.5 with s = 0, u = 0.375, V = 0.375 as t increases
Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes at, + and x at y = 0.5 with s = 0,p = 0375,11 = 0.375 as t
increases .................................
Gauss-Galerkin Finite Element Method: Changes of the “moment”
m§,(y, t) at y = 0.5 with s = 0,11 = 0.375, V = 0.375 as t increases
Gauss-Galerkin Finite Element Method: Changes of the “total mo-
ment” Mf, (t) at y=0.5 with s = 0, p = 0.375, V =‘0.375 as t increases
GaussGalerkin Finite Element Method: Changes of the nodes i, +
andxaty=0.5 withs=2,h=0.5,p=0,u=0astincreases
Gauss-Galerkin Finite Element Method: Changes of the weights at
nodes at, + and x at y = 0.5 with s = 2,h = 0.5,p = 0,11 = 0 as t
increases .................................
Gauss-Galerkin Finite Element Method: Changes of the “moment”
mf,(y, t) at y = 0.5 with s = 2, h = 0.5,p = 0,11 = 0 as t increases
Gauss-Galerkin Finite Element Method: Changes of the “total mo-
ment” Mf, (t) at y=0.5 with s = 2, h = 0.5,;1 = 0,11 = 0 as t increases

141

142

143

144

145

146

147

148

' 6.1

6.2

6.3

6.4

6.6

6.7

LIST OF FIGURES

Gauss-Galerkin Finite Element Method: Movement of the nodes i, +
and x as t increases ............................
Gauss-Galerkin Finite Element Method: Movement of the nodes 0, o,
x, + and e as t increases .........................
Gauss-Galerkin Finite Element Method: Movement of the nodes at, +
and x with p = 2 and q = 2 as t increases ................
Gauss-Galerkin Finite Element Method: Movement of the nodes t, +
and x with p = 1 and q = 2 as t increases ................
Gauss-Galerkin Finite Element Method: Movement of the nodes *, +
and x with p = 2 and q =1 as t increases ................
Gauss-Galerkin Finite Element Method: Movement of the nodes at, +
and x at y = 0.5 with s = 0,11 = 0.375, V = 0.375 as t increases

Gauss-Galerkin Finite Element Method: Movement of the nodes 1:, +
and x at y= 0.5 with s = 2,h= 0.5,u= 0,u= 0 as t increases

xi

134

135

136

137

138

139

140

CHAPTER 1

Introduction

1.1 Background and Motivation

In this dissertation we are concerned with a Gauss-Galerkin finite element method

for a class of singular diffusion equations in two space variables. Let Q = (0,1)2 be

the unit square. We consider the Fokker-Planck equation

Bu _ 3931 162(b2u) _ M 102nm)

 

797 _ 82: 2? 6y 2 6y? (1'1)
with boundary conditions
Bu 2 g on (O,T) x BIZ (1.2)
and initial condition
u(x,y,0) = uo(2:,y) on $2 (1.3)

(1.1 ) describes the probability density u = u(:1:, y, t) for a stochastic process governed

by a set of two stochastic differential equations in (x,y) with the respective drifts

a = a(.7:, t) and c = C(y, t) and diffusions b = b(2:, t) and d = d(y, t):

dY = cdt + ddW2 (1.4)
P{X(0) = 2:, Y(0) = y} = p(:v, y) = given

where W1 = W1(t) and W2 = W2(t) are independent standard Wiener processes. This
is a special case of (2.19) with n = 2 and b12 = b21 = 0. We are concerned with the
case when the problem is “singular” in the x variable and “regular” in the y variable,
i.e., the coefficient b2 may vanish along :1: = 0 and a: = 1, but c and d2 are bounded
away from zero for y E [0, 1].

We shall propose and analyze a numerical method called the “Gauss-Galerkin finite
element method” to solve the initial-boundary value problems for the above Fokker-
Planck equations. The method is a generalization of the one dimensional Gauss-
Galerkin method which was originally proposed by Dawson[1] and further developed
by Hajjafar, Salehi and Yen [4]. We shall briefly describe the one dimensional Gauss-
Galerkin method here. More details can be found in [4]. Consider the stochastic

differential equation

dX = a(X, t)dt + b(X, saw (1.5)

for X = X (t) with the initial condition

X(O) 2 X0 given. (1.6)

The state space is assumed to be a finite interval which we take as [0, 1] and W = W(t)
above is the standard Wiener process. It is assumed that the coefficients a and b in
(1.5) are continuous functions of X and t in [0,1] x [0, T] where T > 0 is a constant.

The initial value X0 in (1.6) is assumed to be a random variable so that the solution

X (t) to (1.5) and (1.6) is a Markov process. It is known that if the law of the process

X (t) has a smooth density u(:z:, t), then u(:r, t) satisfies the Fokker-Planck equation

 

 

Bu _ _ 8(au) 1 62(b2u)
u(:1:, 0) = given, (1.8)

plus boundary conditions at :2: = 0 and a: = 1. The operator L = L(t) is known as the
forward Kolmogorov operator. (1.7) and (1.8) lead to a parabolic initial boundary

value problem for u(a:, t). The formal adjoint L‘ = L“(t) of L above is

and is known as the backward Kolmogorov operator. For each u 2 22(5):) in some
appropriate space V, we may multiply (1.7) by v, integrate with respect to :1: from 0
to 1 and obtain

%(u,v) = (u,L"v), v E V, (1.10)

where (-, ) stands for the L2 inner product. We note that in order to lead to (1.10)
from (1.7) it is necessary that all the boundary terms resulting from integrations by
parts vanish. We are concerned with singular processes for which the coefficients a

and b vanish at boundary :1: = 0 and :c = 1. To solve (1.10), we replace u(a:, t)d:r by

M:

dun($,t) = ak(t)6z,.md:c (1.11)

k 1

associated with an n-point discrete measure pn(t), where in (1.11) :13,c = mk(t), k =
1,2, - . -,n, are the nodes and the ak’s are the corresponding weights. We also let

{u,-(x)},z' = 1, 2, - - - , 2n, be a set of linearly independent functions in V. Substituting

pn(:r,t) and each of v,(a:) into (1.10) yieds

—Zak(t)v,((xkt =20. Lv,(:z:k(t)),z'=1,2,---,2n. (1.12)

This is a system of 2n ordinary differential equations for the n nodes and the n weights
as functions of t. A convergence analysis as n tends to infinity is made in [4]. An
algorithm for computational purpose is also given in [4] along with numerial results
of several test problems. The results show that the Gauss-Galerkin method gives ex-
cellent results and is efficient in capturing the singularities at the boundary for large
t.

As there seems to exist no straightforward way to generalize the one dimensional
Gauss-Galerkin method to a two dimensional one (there is no direct Gauss quadra-
ture formula in two dimensions), Huang and Yen [5] made a modest generalization
by discretizing the partial differential equations by the finite difference method in the
y direction and then solving the resulting systems of partial differential equations in
x and t by the Gauss-Galerkin method. The results showed that the method in [5] is
indeed capable of treating singular parabolic partial differential equations.

In our Gauss-Galerkin finite element method, finite element method is made of the y
variable while one dimensional Gauss-Galerkin method is used for the x variable. The
convergence of the finite element approximation is established first based on the so
called “energy” type estimates. Then, using the theories of measures and moments,
the Gauss-Galerkin approximation for the semi-discrete equations is shown to con-
verge when one dimensional Gauss—Galerkin approximation is made in x direction.
Finally, Combining the above results, the convergence of Gauss-Galerkin finite ele-
ment approximation is established.

We use piecewise linear finite elements in our numerical computations. we first test

a problem where the exact solution is known. By comparing the numerical solution

with the exact solution, we find that the approximation is very efficient and accurate,
even when only a few finite elements and only a few Gauss-Galerkin nodes are used.
For our model problem I Case I and model problem II, we compare the numerical
solutions between our Gauass-Galerkin finite element method and Gauss-Galerkin
finite difference method [5]. Those two methods produce almost identical numerical
solutions. Whereas in [5] it is shown that the Gauss-Galerkin finite difference method
is superior to the traditional two dimensional finite difference method in achieving
high accuracy for solving this type of singular Fokker-Planck equations, Our results
suggest that the Guass-Galerkin finite element method is at least as efficient and
accurate as the Gauss-Galerkin finite difference method. We also study the depen-
dence of the solutions upon parameters in the singularities and the dependency of
the solutions upon lower order terms in our Fokker-Planck equations by applying our
method to several problems. The numerical solutions obtained show that the pro-
posed method is indeed capable in handling initial-boundary problems for singular

diffusion equations of the type considered here.

1.2 Organization of the Dissertation

This dissertation is organized as follows. Chapter 2 contains some basic material
related to stochastic processes and Fokker-Planck equations. Chapter 3 discusses the
problem formulation and some mathematical properties. We set up the Fokker-Planck
equations with a set of boundary conditions. Then we obtain a weak form in the y
variable for fixed x and t. We also obtain several energy estimates. Chapter 4 estab-
lishes the Gauss-Galerkin finite element approximation and includes two parts. One
is the one dimensional finite element approximation in the y variable while the other
the one dimensional Gauss-Galerkin approximation in the x variable. In Chapter 5

we study the convergence of the Gauss-Galerkin finite element approximation. First,

we establish the convergence of the semi-discrete finite element approximation in y.
We then prove the convergence of the Gauss-Galerkin approximation in x. Finally,
by combining the above results, the proof of the convergence of the Gauss-Galerkin
finite element approximation is completed. In Chapter 6 we present our numerial
results for problems involving several partial differential equations and discuss such

numerical results. Chapter 7 contains conclusions and further discussions.

CHAPTER 2

Preliminaries

2. 1 Markov Process

Let T C [0, 00) and BRn be the Borel a—algebra of IR”. Consider a stochastic process
in) = (X1(t),X2(t), - --,X,,(t))T,t e T, defined on a probability space (9, Jr, P).

For all s and t in T such that
0$s<t<+oo, (2.1)

for all :if = (11:1,:c2, - - - ,xn)T 6 IR” and for all B in BRn, we denote the conditional

probability of the event {X(t) E B} given X(s) = a? as P(s, 53'; t, B):

P(s,:i:’; t, B) = rpm) 6 BIXXS) = 2} = /

P(s,;i:’;t,dy’). (2.2)
3768

If the probability P(s, 55'; t, dy') on (IR”, BR") has a density with respect to the Lebesgue

measure dy' , we denote this density as p(s, if; t, 37) such that

P(3,f;t,d37) =p(8,f;t,37)d37 (23)

A stochastic process {X (t),t E T} is called a Markov process if for any finite m and

any collection of times

OSu1<u2<~~<um<s<t (2.4)

"2

in T, we have for all 331,1; ,--«,:i:"" and fin 1R":

“{X()€B|X(u1)=i=‘15W)=i“,-°-,X(um)=fm,f(s)=f}

.. (2.5)
=P{X(t)€B|X(8)=$}= P(8$ t3)
The Markov property (2.5) implies that
P(s.:r:';t,B) = / P(s,f;u,d37)P(u,i/';t, B) (2.6)
Rn

Equation (2.6) is called the Chapman — K olmogorov equation for Markov processes.

2.2 Transition Probability

For all s and all t such that 0 S s < t < 00 and for all B in BRn, P(s, 53'; t, B) is called

a transition probability if
(a) Probability B H P(s, 53'; t, B) is defined for all fixed 3, t and 52'

(b) For all fixed s,t and B, the mapping 53' I—> P(s,:i:'; t, B) is measurable from R"

into [0, 1], and

(c) For all fixed .3, t, a? and B, the conditional probability P satisfies the Chapman-

Kolmogorov equation (2.6).

2.3 Diffusion Process

Markov process {X(t),t E T} is called a diffusion process if

(a) Forallc>0,a:€IR" and0§t<t+hwehave

(b)

1
1' —/ Pt, “;t h,d =0, 2.7
iiiih (Ir—flee ( x + if) ( )

where h I 0 means h —) 0 with positive values, i.e., h > 0.
There exists a function Ei(:i:', t) = (a1(i', t),a2(:i:', t), - - - ,an(§:', t))T such that for

allc>0,:i:'ElR",jE{1,2,---,n} andO§t<t+hwehave

. 1 q _ -
1’53 H (g_f||2€(y, — x,~)P(t, :17, t + h, cry“) _ aJ(a:, t), (2.8)

where ("i is called the drift vector and if n = 1 the drift coefficient.

There exists a matrix function b(§:’, t) = {bjk(f,t)}?,k:1 such that for all e >

0,f€R",j,k€{1,2,~-,n} and0§t<t+hwe have

1

’33 i; [lg—fIIZ€(yj — xj)(yk — mk)P(ti (E; t + h: 6137) = jk(xa t): (2'9)

where for all :i:' and t, the matrix b(§:’, t) is symmetric and positive but not
necessarily positive-definite. The matrix b(:i:‘, t) is called the diffusion matrix

and if n = 1 the diffusion coefficient.

10
2.4 Fokker-Planck Equation for a Diffusion Pro-
cess

Let X(t) = (X1(t),X2(t), - - - ,Xn(t))T,t E T be a Markov process. We assume that

the process X (t) satisfies the following three hypotheses (Schuss [9] and Soize [11]).

Hypothesis 2.4.1 The transition probability P(s, f; t, dy) on (Rn, BR") has a density

p(s,:i:'; t, y) with respect to the Lebesgue measure (137 on IR" such that
1309,55; t, (137) = 10(8, :3; t, 3716137 (2-10)

12$P(s,f;t+ h, B) 2 13(52'), VB 6 BRn. (2.11)

Hypothesis 2.4.2 The mar/coy process X is a diffusion process. Consequently, the
process X has almost surely continuous trajectories and equations (2.7), (2.8) and
(2.9) hold with (2.10). We assume in addition that equations (2.7), (2.8) and (2.9)
are satisfied uniformly with respect to :3. For all j, k E {1, 2, ~ . . , n} and 0 _<_ t < t+h,

we then have

1
‘ “'73 h, C”=0, 2.12
iiiih IIJ—f||26p(,$’ + if) y ( l
1 1 ~_ #
iii)! 3 IIfi—fl|2e(yj _ lepu’ ‘7” t + hit/7‘11! - 0195,15), (2-13)
1
’ " ' - “' ”= - “ 2.14
fig), “1,4”ij inky]. xk)p(t,:c,t+h,z7)dy b,,.(x,t), ( )

where d(:i:', t) is the drift vector and b(:i:', t) the diffusion matrix.

a
Hypothesis 2.4.3 Functions d(;i:', t), b(1':', t), %p(s,f;t,y'), 5—(aj(y',t)p(s,f;t,y')),
62

,—b- “,t s,:i:';t, ,V°,k E 1,2,---,n , exist and are
ayjaykhdy )p( 37)) J { }

J

Vj E {1,2,---,n}

continuous.

11

Theorem 2.4.1 (Fokker-Planck equation for the transition probability). Under the
hypotheses 2.4.1, 2.4.2 and 2.4.3 and for all fixed 3 Z 0 and f 6 1R", p(s,:i:';t,y')
of the diffusion process X (t) satisfies the Fokker-Planck equation (or the forward

K olmogorov equation)

_, 1 n 32 _. .._
—(aj(y,t)P(8,$,ta§)) + 5 g; mwjdyfllflsflm 31))- (2-15)

Theorem 2.4.2 (Fokker-Planck equation for the probability density function) Let

X (t) be a diffusion process satisfying hypothesis 2.4.1, 2.4.2 and 2.4.3, and such that

—o

22(0) = X0 as, (2.16)
where X0 is a random variable with probability law

Pr,(d17) = prohfidfl, (2.17)

where p30 is a given continuous probability density function on IR" with respect to the
Lebesgue measure dy'. Then, for all fixed t E T, the probability law P in)“, dy') of
X (t) is written as

Prawn dz?) = Mt, W27, (2.18)

where the probability density function pg(t,1]) on IR" with respect to (it? satisfies the

Fokker-Planck equation
6pX°_ " 8 1 _,

— — — — —— b- t ~ t . 2.19

Zlay (aj( ytt )px(t aml-f— 2 Z ayjayk(1k(yi )pX( 137)) ( )

J

12

with the initial condition

Px'(0, 37) = WOW), V37 6 IR". (220)

2.5 Stochastic Differential Equation

We denote by L£[a,fl], (1 S p < 00), the class of all non-anticipative stochastic
processes f (t) satisfying:
3
P{/ lf(t)|”dt<oo} :1.

We denote by Mafia, 6], (1 S p < 00), the subset of Lacy, 6] consisting of all stochastic

processes f(t) with

Rt]: |f(t)|”dt} < oo.

The stochastic differential equation
dX(t) = sum), t)dt + b(X(t), t)dl/V(t), (2.21)
is defined by It?) integral equation
X(t) = 56(0) + /01 (202(3), s)ds + [0‘ b(X(s), s)dl/V(s), (2.22)

where W(t) = (W1(t), W2(t), - - -,W,,(t))T is a vector of independent Brownian mo-

tion.

13

2.6 Existence and Uniqueness

Let [:3]2 = Z x? and |b|2 = Z ijklz. We consider the stochastic differential equation
j=1 j,k=1

(12(1) = with), t)dt + b(X(t), t)dl/V(t), (2.23)

56(0) = X}, as (2.24)

Theorem 2.6.1 Suppose 6(52', t) E Lgy[a,)6] and b(.'i:', t) E L3,]a,fl] are measurable in

(53', t) 6 R" x [0,T] and there exist constants K and K. such that
|5(f,t)| S K(1+ lfl), |b(f,t)| S K(1+ lfl), (2-25)

and

(r, t) — as, t)| g K...|:i:‘ — 53, (as, t) — 5(55, t)| g K..|:i:‘ — a (2.26)

for f 6 IR" and 0 S t S T. Let X0 be any n-dimensional random variable independent
ofo —algebra f(VI/(t), 0 S t S T) such that ElXiol2 < 00. Then there exists a unique

solution X(t) of {2.23) and (2.24) in M2,, [(1, s] (Friedman[3]).

CHAPTER 3

Problem Formulation and

Mathematical Properties

3.1 Initial-Boundary Value Problems and Their

Weak Formulation

Let Q = (0,1)2 be the unit square, [0,T], T > 0 be a time interval of interest,
9 = g(x, y, t) and uo = uo(x, y) 2 0 be given data. We consider the following Fokker-

Planck equation

Bu __ _M 182(b2u) _ 8_(c_u_) +_1_(92(d2u) 1

‘5?" 0x 2 6x2 6y 2 6y? n (O’Tlxn (3'1)

with boundary conditions
Bill. 2 g,- 011 (0,T) X 69 (Z = 1, 2,3,4) (3.2)

and initial condition

u(x,y,0) = uo(x,y) on 9 (3.3)

14

15

(3.1 ) describes the probability density u = u(x, y, t) for a stochastic process governed
by a set of two stochastic differential equations in (x,y) with the respective drifts

a = a(x, t) and c = c(y, t) and diffusions b = b(x, t) and d = d(y, t):

dX _—. adt + del
dY = cdt + (1sz (3-4)
P{X(0) = 27, WW = y} = p(rr, v) = given

where W1 = W1(t) and W2 = W2(t) are independent standard Wiener processes.

We note that in the more general case with drift vector ('1' = (a1, a2)T and diffusion ma-

b11 12
trix b = that depend on x, y and t, the stochastic differential equations

(’21 b22
are given by the following:

dX : aldt + blldWI + b12dW2
dY = 0.th + bgldW1 'I" b22dW2 (35)
P{X(0) = I. Y(0) = y} = p(x, y) = given-

The probability density function p(x, y, t) is governed by the following (Schuss [9]):

3P _ a(all?) 3(a2p)+_1_02(a11p) 16201121?) 16201211?) 302612219)

at 62: By 2 662 2 6x03} 2 Byax + 2 83/2
(3.6)

where

aij = (bbT)ij Z,j = 1, 2.

In the particular case, a1 = a1(x,t), a2 = a2(y, t), bu = b11(x,t), by = b22(y,t) and
b12 = b21 = 0, the above equation (3.6) reduces to (3.1) with p being replaced by u,
a1 being replaced by a, a2 being replaced by c, bu being replaced by b and b22 being

replaced by d. Henceforth, we shall be primarily concerned with (3.1).

16

We assume that a, b, c and d are smooth functions with uniformly bounded derivatives
of all orders. Many significant results in solving parabolic problems and efficient
numerical methods are obtained by Luskin and Rannacher [6], Quarteroni and Valli
[8], Strang and Fix [13], Thomee [14], [15], [16], Stoer and Bulirsch [12]. In this
dissertation, we shall be concerned with the case when the problem is “singular”.
More specifically, we are concerned with the case when the problem is “singular” in
the x variable and “regular” in the y variable, i.e., the coefficient b2 may vanish along
x = 0 and x = 1, but c and d2 are bounded away from zero for y E [0, 1].

For boundary conditions in the y-direction we consider along y = 0 the Dirichlet
condition

B3u = u(x, 0, t) = 0 along y = 0 (3.7)
or the Neumann condition

6u(x, 0, t)

B =
3” 8y

2 0 along y = 0 (3.8)

Similarly along y = 1 we consider the Dirichlet condition

B4u = u(x, 1, t) = 0 along y = 1 (3.9)

or the Neumann condition

8u(x, 1, t)

B =
4’“ 6y

= 0 along y = 1 (3.10)

The boundary conditions Blu = 0 and Bgu = 0 depend on the type of singularity

one has at x = 0 and x = 1. Suppose that

b2(x,t) = 0(x”) as x —> 0 (3.11)

17

b2(x,t) = 0((1— x)q) as x —> 1

(3.12)

where p and q are constants satisfying p Z 1 and q 2 1. The appropriate boundary

conditions, according to Feller [2] in his study of one-dimensional singular stochastic

problems on (0, T) x (0,1), Keller and Voronka [7], are

 

 

 

 

Blu=u(0,y,t)=0 if p>1
Bluzlgxuzo if p=1 and
or
, 16(b2u) . _
Blu—alcign){au—-2- 6x }—0 1f p—l and
and
Bgu=u(1,y,t)=0 if q>1
Bw==li13(1—x)u=0 if q=1 and
or
, 16(b2u) , _
Bgu—algr}{au—§ 8x }—0 1f q—l and
We now consider the following initial-boundary problem:
. a—u—O+Lu f in (or)xa
at — . I
u,(x 0 ,=t) on (0 T) x (0,1)
u,(x 1 t,)=0 on (0 T)x(0,1)
Blu(0, y, t) = 0 on (0 T) x (0,1)
B2u(1,y,t) = 0 on (0 T) x (0,1)
u(x, y, 0) = uo on Q
where
L _ 8(au) _162(b2u) 6(cu)_1 32(d2u )
u — 3x 2 8x2 8y _2 By?

 

a(0, y, t) = O

0(0, y, t) as 0
(3.13)

a(1, y, t) = 0

00.11.10 9* 0
(3.14)
(3.15)
(3.16)

18

It is seen that for simplicity in the subsequenct analysis we have assumed Dirichlet
conditions at y = 0 and y = 1. Problems with Neumann conditions at y = 0 and y = 1
require only minor modifications. Also we have included a non homogeneous term
f = f (x, y, t) in the partial differential equation. Let V = HMO, 1). Multiplying both
sides of the partial differential equation in (3.15) by v = v(y) E V and integrating

over (0,1), we get

 

 

 

 

    

 

         

 

 

 

 

 

/.‘Z.—“ Mi.“— . — 44/323222
(3.17)
—/1162é::u)vdy=/1fvdy.
0 98-?” 16(80. vd—y [01 la2;::u)vd y+/o1 g—Suvdy+/Olcg—:vdy
.1- — . 3 22—; :63) y_ :=/
Since vlyzo = 0, v|y=1 = 0, we get
(3.18)
+/olcg—:-v dy :olégglu%dy+l 0 2 d%:%:dy =/0 fvdy.
Let the bilinear form a(-, ) be defined by
a(u,v) = [01 6S:)vdy— 01%62;::u)vdy+/01 535mm?)
(3.19)

+AICQS-Wd +Alé%€lugflw +/02d2c9y 23_UZU@

19
a(u,v) = (Bight) — (é 62;::“),v) + (3311,11)

By’ 2 8y ’6y 2 By’ay ’

1
where (f, g) = [0 f(x,y,t)g(:r,y,t)dy is the scalar product of f and g in L2(0, 1)

 

 

(3.20)

with respect to variable y. We also denote the norm of f E L2(O, 1) with respect to
1

y by ||f(x,-,t)ll3 = [0 |f(:c,y.t)|2dy

The weak formulation of (3.15) now reads as follows: Given f and no, find 11 such

that
(114,12) + a(u,v) = (f, 12) V1) 6 V

< (3.21)

u(x, y, 0) = uo.

 

We note here that if homogeneous Neumann conditions are posed,the weak formu-
lation above is to be modified with the space HMO, 1) replaced by H1(0, 1) as such

Neumann conditions are “natural” boundary conditions.

3.2 Mathematical Analysis of Initial-Boundary

Value Problem

Let the bilinear form b(-, .) be defined by

b(u, v) = a(u, v) — (1(1), 11) (3.22)

20

Taking 'u = u in (3.20), we have

__ 6(au) 182(b2u) ac Bu
a(u,u) —— ( 6:1: ,u) — (5 82:2 ,u) + (53711, u) + (055,11)

2 8y ’8y 2 By’ay

Taking v = 11., in (3.22), we obtain

 

 

(3.23)

b(u, at) = a(u, ut) — a(ut, u)
3(au) u _ 102(b2u) u + 26-11 11
3:1: 1 t 2 61:2 a t 0y 7 t
8(aut) 182 (b221,) go
( 0:1: “) (2 (9:02 ’“ + ayut’“

ay’ 2 6y "ay 2 ay’ay '

 

 

 

 

 

Letting

 

....) z (a<;;u>,.) _. (21.122...) . (_.,.)

+ 93 + 16(d2)tu6_u + _1_(d2)8_u it)
“ay’” 2 8y ’6y 2 ‘ay’ay

(3.25)

 

we have

 

 

1

21

) 1 62 ((122)
,u) (2— 8x2

 

 

0(d2).
011

 

 

 

6(d2)
73?

0(atu)
8:1:

 

"(Z-Mb
.) _ (_
....) G)
2).)H
)—) (

1
2

1
2

62((b2)u,)
63:2

02((b2)U)

02:2 ’

_ l 62( ()b2 ))t’U,
,u 2 012:2

2811. 3__ut
2 (9y 0_y

 

))’U.t 1 62“
’ u 2 5:2)

9M
3%))
29+)

9 2U)ut)

u’ay 2 59y“ 3y
1 2011) (911 1 2611 But
2“l M:y’ay) + (2‘1 6y 8y)

,)-)(———)2<1;))

u + @1111 + c9211
1 6y: taya

~22)»

+Euu+c
7“ 8y“

6126—1“ (911
a_y B—y

...) . (<1...) .. (.
l

——,u

V\_l_/

0

Bu
8

ant )
_, u
y

_a at)
y

22

 

10(d2)¢ Bu 1 2 Bu Bu
('2' 6y “’a)+(é‘d”a’a)}
6(au) 1 02((b2)u) ac Bu
2{( a) Wt) “ (577)”) + (5?“ 1“) + (CW)
2 0y)“ ’20 (2 371’???»
0(aut) 182( ()b2 )u, But
{(7 ) (2i— ax) )“)+(§§“~"+) (Cw)
+ Gag: __ u 126—2“ Bu
)ut’afl) (2d 07’ ’02)}

 

+

 

(’21?) ) 6%)“ )+(-W)+(c§%e)

10(d7)u %_ut 612,611, 621—:
2 Oyu 2d 0y 6y

So
d
flaw, u) = at(u, u) + 2a(u, ut) — b(u, at). (3.26)
Thus we have
1 1 1
a(u,ut) = Eat-a(u, u) — iat(u, u) + §b(u, ut) (3.27)

We now ssume that a(-, ) is continuous and coercive, i.e., there is a positive constant

a, independent of t, such that

a(v,v) _>_ allvllf, V1) 6 V. (3.28)

Theorem 3.2.1 If no 6 L2(O,1),f E L2(0, 1), then for almost every :1: 6 (0,1), the

23

energy estimate

0<t<T

1 T
s Hum-m3 + 2; f0 Ilf(z,-,t)||3dt

T
munm,,mt+a/ImuWUMm
0 (3.29)

holds.

Proof. By (3.21), we have

(ut, v) + a(u, v) = (f,v)

Taking v = u and using (3.28), we get

(uhu) + a(u,u) = (f, u)

3%(u H)+MUW (in

ignw( mt+amno=tnm

11mm, mus + aIIu(z,-,tm% s §%11u(x,-,t>113+ a(mu)
:HUMHS%flfl%nM%+;W@mfl%

1 (1

Integrating over (0, 7'), T E (0, T], we obtain

1 Td T
E/Eflm($mfl%fi+afllmfinflmfi
a T
:1:-,, d —/0 ,-,t2dt
_Ya/an MHt+2 ”Mm m
5mmwm—wMW 0)It+ [HuHI%S;foHHM
lWWwflfi+aAlWWmflmflSH%@r)m+EAflU($nt)%fi

So,

T 1 T
FMXW($HQ%+QAlWWmflmfiSWM%J%+EAlUWmflMfl D

0<t<T

24

From now on, we shall assume that

la(u,v)l S CHWL‘, -,t)||1||v($, -It)||1
lat(’u,v)l S CHUCK, °,t)l|1||v($, -,t)l|1 (3-30)
|b(u,v)| S Cllucva -,t)||1||v(:v, -,t)||o

Theorem 3.2.2 If no 6 H6(O,1),f E L2(O, 1), then for almost every 3: 6 (0,1) the

energy estimate

514$ 39w,
sup ”u(x, t)lli+ if II ———93tII3dt

te(0,T)

(3.31)
SCa{lluo(, )Ili+/ ||f( mange}

holds, where Co > 0 is a constant independent of T.

Proof. By (3.21),
(uhv) + a(u,v) :- (f, 1)).

Taking v = at, we obtain

(abut) + a(u, at) = (f, at)

“WHO +a(u, Ut): (f Ht)-

By (3.27), we obtain

lat(u,u) + %b(u,ut) = (f, at).

a(u, u) — 2

1
2 -—

25

So,

(u,u) = é—at(a,u) — %b(u, at) + (f, at)

C C
5IIuII3 + 5HU|I1|IIIz||o+l<qut>|~

1
Hutllgi’ 237a

|/\

Integrating over (0, t),t E (O, T], we obtain

[IIusII3ds+5/‘js —aquId
t
55/ IIuII3ds+—/ IIIIIllllusllod8+f IImsIIds
o 2 o o

t t 1 t 1 t t
36/0 IIuII3ds+c2/O IIuII3ds+5/O IIusII3ds+5/O llusll3d8+/O IIfII3ds.
Hence,
1/‘II ||2ds+lftd ( )d8 < cI1+cI/‘IIIII2ds+/‘IIIII2ds
— s — —a u,u _ I -
2 o u 0 2 0 ds 0 1 o 0

By (3.29), we have

fIIuII3ds<—51,-IIuoII3+ fi/ IIfII3ds.

 

 

 

           

 

So,
1/ ‘ llusllfids + 5aIuItI, u(t)) — 5aIuI0I, IIOII
sc<1+c)IIUoll3+ (1+4 C))/0T”f“3d3
5/Hmna+5MWIIumI
$1 01% +1+(1+C))/0Tllf||§d&

5a (u (0) u(0))+

26

By assumption of (3.28) and (3.30), we have

a(U(t)IU(t))ZallIIlli, a(U(0)IU(0))SCIIUOIIi and Huollgflluolli-

Therefore

 

 

c(1+c)
/5||u.||3ds+5 I|u||3_ 5I|u uo||3+ || uo||3+

+c T
))/ I|f||3ds
0
1 1+c 2 c1+c T
IIu|I3+— ;/|Ius||3ds <—(5+ a )|Iuo||3+5(I+ ( ))/ ||f||3ds
0

2 8n 2 2 T 2
sup ||u|| +51] ll—llodt<0 ||u0I|1+/0 ||f||od3 III

t€(0T),

 

 

A simple consequence of Theorem (3.2.2) is given by

Corollary 3.2.1 Assume that for almost every :2: 6 (0,1) andt E [0, T], u in Theo-

rem 3. 2. 2 satisfies

IIU($I -It)||§ S C(IILUWI °,t)||3 + “WEI °,t)lli)- (3-32)

   

 

 

Then for almost every x 6 (0,1) it satisfies the estimate
2

2+
£3331”qu W’t)” Eli/OT{6L($8,T— 0

(3.33)
s 0.. (Hugo, -III3 + [0 ”f(x, at-,:II|3dt}

Proof. Estimate (3.33) follows at once from (3.32), (3.31) and (3.29), since Lu :
8n
f — 55.

CHAPTER 4

Gauss-Galerkin Finite Element

Method

4.1 Finite Element Approximation in the y Vari-
able

Let Vh C H3(0, 1) be a finite dimensional subspace and {¢I(y)}f:”0 be base functions

of Vh. By (3.18), we have

 

 

     

   

16a 10(au) l132(b
0 6t vyd H/o—di—vdy o 2 8:5 2 (ml/0133’ Cuvdy
4.1)
1 8u1d21(0 d28_u6_v_d (
+/0 candy+ 0 ~2—-(—y—ug — dy+/12: y—/:.fvdy
Taking v = ¢I(y),l = O, 1, - - -,Ny, we obtain
‘35qu 16;“ (ydy) —1162(b2“)dI Id
0 z( 0 2 8:1: 2 (y y
116d?
+/g —u¢IIyIdy+/?x c—IIIIII IIId +/5 -(—y ) udIIydyI (4.2)

+/5d1 d3g—yudIII/I dy— / deIyIdy

27

28

We approximate u(x, y, t) by

 

“(I’M/t'V Zaj($It)¢j(y-) (4-3)
From (4.2), we obtain
Mao-(Mt 16(a()aa:t)
J l 3 )d __ _ __.7_______(
go— )/ My )My )3! Z/( Mindy
Nv 1 2 2a. :1) C

+5121) 6 (b53315 ’t))¢z(y)¢j(y)dy-N 201ml?) 0 (ligwywyywy

N” 16dz( ) (4.4)
~20aI-IdtI [0 chIII ~2aII :rt)/ —dIIy Id dI-IyIdII

or,

1 N5, 1 1
—§gaj(x,t) [0 33¢;(y)¢;(y)dy+ f0 de(y)dy

 

 

 

 

 

( Bao($,t) \

6018(2),”
(WIO) dIIIII dIIINII) ‘5‘

Ba 11:,t

I——-—”3§ ))
_ 23/1333,“ ——-——-dIIyIdI-IyIdy

(4.5)

+§§Ala(bg;§$’ t))¢,y()¢,-(y) )—dy gay-(cam %¢I(y)¢j(y)dy

NV 1 N1, 18
‘gaihhtl f0 C¢I(y)¢3(y)dy - 520301315) [0 iayl¢i(y)¢j(y)dy

3:0

1 NV 1 2 I I 1
71235.33) [0 d ¢,(y)¢,(y)dy+ f0 deIyIdy,

29

where day) = [01¢z(y)¢j(y)dy- Thus

( 43(0, 0) 45(0, 1)

¢(1I0) ¢(1,1)

 

{Z/laaa,(xt

=_ Z/Iama,“

 

N" 13(aaj(ddI :))
KEEI/O 3:1:

( .1.
25:0

1N

\ d(N,,,0) ¢IN5,1)

 

 

W W

 

(I’M/(3} )ij(3/ )dy /

”v 1 2 20.3:
Z/o a (b 3;: ,t))¢0(y)¢j(y)dy

, l32(‘52032'(117It))
+ ‘ [I

2 j=0

1NI

 

1 82(b20j(.’13,t))

39:2 ¢1(y)¢j(y)dy

 

Kig/O 83:2

¢NI (y)¢j(y)

( 300 (2:, t)
at

 

601(1), t)
at

 

 

\

 

dy

/

 

W

And

30

 

 

Nv . t 1 6C d
(12:30:03, ) 0 a—y¢N,(3/)¢j(y) y)

{£255 -/01§£d2—¢o(y)¢- dy\

i=0

52501138 a“? —-)y¢'1(y )¢I(y)dy

 

 

52C” j/olaécf) GSA/"(3! )¢j(y 30d?!)

( [01 f¢o(y)dy )

[0‘ f¢o(y)dy

 

 

( fol f¢o(y)dy )

N, 1
( gajmtvo C¢o(y)¢}(y)dy \

N5 1
EGAN) f0 c¢1(y)¢3(y)dy

 

 

( :aI-Ix, t) [01 cd~.IyId;IyIdy )
( liaj/o d2¢0(y) y)

1%: '/1d2¢’()M)d
23:00] 0 1y y y

1"“

 

 

(gzdI/O d3dII,IyIdI I y,

31

 

 

 

 

 

 

( acreage,” \ f if; memm \
23% = _qu 23/1 a( (a___a__J( x, t)) wag-(may
W / K Jill W:~:(y)¢j(y)dy /

f g]: [01 ‘92“:9 t))¢o(y)¢j(y)dy \

 

1 1 NU 102 2 . IE,
+<I>- —2 [O (b 3;: t))¢1(y)¢j(y)dy

 

 

 

1 N” 1 82 201' a:
\ 5&1) (b 6;: ’t))¢~y(y)¢j(y)dy)

N11 1 C
( 20‘2” t)/0(9 8—y¢o(y MHZ/My \

_<p—-1 2am, t) / g—gaslwm-(wdy

 

 

_Q-l

_Q-l

+<I>-1

Define

32

(”201(1513/0 C¢o(y)¢3(y)dy \

NV 1
gag-(m) [0 c¢l(y)¢3(y)dy

 

 

\§,01((xt>>/ c¢~y<y)¢;(y)dy )

1”” 1 d2
(52%A62W)<MAW) \

i=0

 

 

 

 

 

 

5:“ flaédj)¢i(m )¢J(ydy) —¢—1
"2:01 flag? —-y-)¢’N,,(y )¢j(y y)dy)
( [Ma/MN
folquamdy
\ A1f¢;(y)dy }

Mimi):(ao(a:,t),a1(x,t),...

 

’ 0N1: (:13, t))T

 

(4.8)

33

and let L1 be defined by (4.7), we can write (4.7) as

 

 

 

6&($,t) _ -
at — Lla(z, t) (4'9)
Using (4.3), we obtain
/ 1u(z,y 0)¢z(y)dy
= [01:03-03 0)¢j(y)¢z(y)dy
(4.10)
— 2:;an 0) fol ¢z(y)¢.(y)dy
— joint)“; J)
That is,
{ a0(x,0) \
1 a1 51:,0
f0 a(x,y,0)¢z(y)dy=(¢(1,0) ¢(z,1) ¢(1,Ny)) (_ )
KaNy($,0) )
( Alu(x,y,0)¢o(y)dy \ { ¢(0,0) ¢(0,1) ¢(0,Ny) \ f a0(:z:,0) \
AIU($,y,0)¢1(3/)dy : ¢(1,0) ¢(1,1) ¢(1,Ny) a1(a:,0)
\AIU($,y,0)¢~.(y)d3// \¢(Ny,0) ¢(Ny,1) ¢(Ny,Ny)) \GNJxfiU

 

 

 

 

 

 

(4.11)

34

( crow) ) ( [Glu(x,y.0)¢o<y>dy )
[01u(x.y,0>¢1(y)dy

al (1:, O)

= <I>‘1 (4.12)

 

 

 

 

(awn) ) ( /1u(x.y,o>¢~.<y>dy )
4.2 Gauss-Galerkin Approximation in the x Vari-

able
We approximate aj(a:, t)da: by
dpj( x, t) 2:21wij(t) —a:,-J-)(t)d:1:, j: 0,1,~-,Ny, (4.13)

i.e., each aJ-(x, t) is associated with an n-point discrete Dirac-delta function, or “mea-
sure”, xij (t), 1 S i g n, are the n “nodes” and wij(t), 1 _<_ i g n, are the corresponding
“weights”. We choose :13", 0 g k 3 2n -— 1 as test functions.

Multiplying by 33", k = 0,1,-- - , 2n — 1, and integrating over [0, 1] in (4.2) we obtain

[01 folat B_U¢M(y) xkdxdy +/01/01 aglxunifiy )3::1;-—kcl:2:d—/Ol [01- 1 82;::1‘)q51( (yx) kdxdy

+/01/01 3511chxkdasdy+/O/00—¢z(y)xykdxd +/0 [0218ng Wkdflidy

 

 

1 11 28a , k _ 1 1 k
+/., lo 5" 5§¢‘(y)“’ dxdy -/0 [0 My” dmdy (414)

35

Using integration by parts, we obtain

[01 [0181: @WW) +/01 [01 gSu¢¢(y)dxdy + [01 [)1 c%¢)(y)d$dy

+/01/0116—(a——:2)( u¢z(y )ydscd +/011/02d2—-¢,(y )dxdy (4.15)

+/01 {an — _3(§:“)}:

 

y)dy =/01/01 f¢z(y)drvdy ('6 = 0)-

 

=0

and

[IA1—¢lxkd$dy—[)1/()Ukauxk1¢ldzdy—/01/01;kb2u$k—2¢ld$dy
+/01/01—u¢z(y) xkdzdy+/0 [0 c—¢)(y) )xykdxd +/0/01§18__(5_d2 y)xkd$dy

1

¢)(y)dy

1:20

1 2U
+/0101d2-—y-¢l(y)$kd$dy+/l{augjk_ giggx k+kb2u$k_1y}

 

= [01 [0‘ 141(4)::skdxdy (k 2 1).
(4.16)

By the assumed boundary condition (3.13) and (3.14), the boundary terms above
drop out in all except the special case when p = q = 1 and a = 0 at x = O and
:1: = 1. We shall assume in the subsequent analysis that we are not in this special
case. Problems in which

special case occurs will be handled separately. (See, e.g., the test problem in

Section 6.2).

36

Thus, we have

[01 [01 gtgdhdxdy + [01 [01 guffizdxdy +1: [01 c-Z—Zqfizdcrdy

+//lla-(—d2) u¢ldx dy +/ / Edz—qbfdxdyzfolfolfgbldxdy (k=0)
(4.17)

and

[0 l0 g—t—qfizxkdxdy— [0 [01 kaux" l¢ldxdy— f0 folék bz—umk 2‘15: dxdy
+/01/0 —u¢¢xkdxdy+/l [01 c—¢zxkdzdy+/01/Olgagf)u¢ixkdxdy

1 11 26—11. I k _ 1 1 k
+/0 0 2d 6y¢‘(y)$ divdy—fo [0 fable/)2: dxdy (1:21).

 

(4.18)
We get the following by using (4.3), (4.18) and (4.17),

Nvdl k 1
Zaowmmmi4wwwm

.40

=2;/ / kanw 1"" 1¢I¢dedy+2/ [lék 1,0,3; as" 2414.41.44.21
/f&=”@WJ“WyZ//%anw)%m
2,] [11659012 5.424. 101wa] [1; 4243414442244

+/01/01 f¢z(y)-’Ekdxdy (k 2 0).
(4.19)

From (4.19) we obtain

(40,0) 40.1) 40,42»)

:22] / kaaja: 2:" 14,4,dzdy+Z/O [1%]:

37

{ dilt— 01a0(x,t)ackd2: \

g-/lo.r(31: t):rkdr
dt 0 1 ’ ‘

 

 

if (Nd
\dtoaw’ 3’ 3’}

bzozjx :r" 2¢1¢jdxdy

—:/01/01‘6_ya BC Wxtfibt “WU/)3; kdxdy— Z/Ol [1 CCYj($t);t/¢1( )¢jy( )1: kdxdy

HI“? My

+ [01 fol f¢z(y)y’°d$dy-

or,

( 440,0) 440,1)

¢(1,0) (15(1, 1)

 

K (b(NyaO) ¢(Ny11) 9M

y)a:—kda:dy 2/0 f0 2dzaj¢,()

¢(0,Ny) ){d/O ao(x,t)a:dx\

d

 

 

d 1 k
32/0 al(:c,t)a: d3:

Ny,Ny) ) \a/OIQNJatflkdx)

M )1? "dydy

 

(4.20)

38

N1 1 1
( gfo f0 may”,tlxk’1¢o(y)¢j(y)dydy \

N1 1 1 k
2 f0 [0 Mag-(my: -2¢1(y)4j(y)da:dy

 

 

\:/01A kan-((:1:t)k—1¢Ny( y)¢](y)d1:dy)

 

(Z/O [011k k-l )b201(($ W k 2(150(y)<?5j(y)d:wly )
23/ [911—1121111441 241(1 )¢j(y)dxdy
(22/ [01%]. k-11b2a:(a: we: 2 2:14~.( >414 mdxdy)

 

/ [186a] (,)(xtqboy ) kdxdy \

fin/0 [DIS—Z: (1:, t) )¢1( (y)1: kdxdy

 

 

)2]; lolgcafl (,)(xthNyy )(bjg/()1:kd1:dy}

 

 

 

39

f/ca,1t¢0 W5)kd$dy\

Nu

23/01/01ch t)¢1(y)¢ ( )y *dydy

i=0

 

”11 1 1
\2/0 [0 caj(1:,t)¢Ny(y)¢;-(y)xkd12dyj

 

f/Haf) W<MAw1mwy)
22/ [Hi—2% ya(y)My )x kdydy
23/ / fl;- t)M,,<y )¢y(y)x’°dccdy /

{ Nu 1 ll .1 ,
321/0 [0 2dzay¢o¢j$kdmdy\ ( A1A1f¢0(y)xkdrydy)

N” 1 l1 ,, 1 1
2:3]0 f0 gdzayylyjykdydy + [0 [0 f¢1(y)1:kd1:dy

 

 

 

(2],) [01; -d2a,-¢’N gb'xkdxdy ) \ fol [01 f¢~,(y)y“dxdy )

40

Thus,

{ Hit/01 ao(x,t)xkda: \

i/lau‘tflkd
dt 0 1 ’ :1:

 

 

d 1 . k
\E/o aNJ(3:,t)a: d1: J
{ 22/01/01 ““103,t)$k‘l¢o(y)¢j(y)d$dy \

Nu

=‘P—l Jag/0 f0 1““!ijt)$k_1¢1(y)¢j(y)dxdy (4.21)

 

 

. f/kanxt 1¢N,,( y)¢J-(y dedy)
{Ea/01 [012k k—1)b2aJ-(:1:,t)xk 2¢0(y )¢j(y)dxdy \

+<I>’1 23/0 A1k2k—1)b2aJ-(x,t)xk 2q§1(y )qu-(y)d:cdy

 

 

\2/0 [011162 k—1)b2ch-(:c,):ct k 2q>1vy(y )q’JJ-(y y)d:z:dy}

 

 

 

NV 1 1 c -
@3/0 [0 g—yajcc,t)¢~y(y)¢j(y)xkdxdy )

Nv

i=0

Z//11L2:%M)¢J(y)xkdxdy\
->:/0/116——y—§,)a-(:c '(ymmfldmy

-Z/ [16 2 (3% a3(x,,,t)¢>'N( (wa-(ywdzdy )

ELI/0100““? t)¢1(y )053 (y )x kdxdy

41
2:] [0130 --aj( :13 t)¢o(y )qu (30$ dxdy \

1:] [01660014 (,2: t) )(gbly (ycc) "dccdy

 

:2]- /0 ca,-( :13, t) )(qfioy y)cckd:cdy \

 

:30]; fol caj( cc, t) )(ngyy (ycc) cckdccdy)

 

 

42
Nu 1 11
{ gfo f0 561201166,t)¢6(y)¢}(y)xkdivdy \

NV 1 11
zgfo /0 §d2aj($,t)¢;(y)¢;(y)xkdmy
J:

 

 

NV 1 11 2 I I
\fg/O [0 5d aj($,t)¢~y(y)¢j(y)$kdxdyj

( fol/01 f¢o(y)xkdxdy \

1 1
+<I>_1 /0/0f¢I(y)xkd:rdy

 

 

\ [(11/()lf¢Ny(3/)$kd$dy /

 

 

 

 

where
/ MD) MI) ¢(0,Ny) \
(I): we) can) ¢(1,N,,)
\¢(Ny10) ¢(Ny,1) ¢(NyaNy) )
and
{ $(0,0) $(0,1) $(0,Ny) \
(p4 = 50,0) 60,1) $0,114,)
\‘flNz/ao) a(Nwl) $(Ny’Ny) )

 

Let L; be the formal adjoint operator of L1 given by (4.21), we can write (4.21) as

d

Emu, t),a:k) = [0 (&($, t), LI(2:k))- (422)

43
From (4.13) and (4.21), we have

( $521.11“th—xio(t))xkdx )

dEt-fol :wil(t)6(x — xil(t))xkd:1:

 

 

d 1 n k
(a; [0 gwiNy(t)6(x—xmy(t))x da: }

”v 1 1 n
{ 2/0 [0 kagwifltMW—3713‘(t))$k_l¢o(y)¢j(y)d$dy \

NV 1 1 n

= (9—1 23/ / ka2w1j(t)6(:r — $ij(t))$k-l¢1(y)¢j(y)d$dy

0 0 i=1

 

 

szg/O f0 kagwifltww —$ij(t))$k_l¢Ny(y)¢j(y)d$dy /
NV 1 1 n
f 211/0 f0 $14k — 1)b2;wij(t)6($ - $11(t))$k-2¢o(y)¢j(y)d$dy \

Nu 1 11 n
+<I>-1 23/0 [0 319(k-1)b2§wg(t)5($-arij(t))x’““2¢1(y)¢j(y)d$dy

 

 

N1; 1 1 n
(go/0 f0 “$1109“1)b2;wij(t)5($—$ij(t))$k"2¢Ny(y)¢j(y)d$dy )

44

 

{ :AIAI-g—Sgwu(t)6(x

_<1>—1 i/olfggwuwa
Nu 1 1 n

1,424) /0 g—zjgwij(t)6(x

 

— wig-(wwoowJ-(kadxdy )
— x11<t1>¢1(y)¢j(y)xkdxdy

- xij(t))¢~,, (y)¢j(y)m’°dxdy )
—xu~( 111101.21 >¢gy< ) *dzdy

_mij“ ))¢1(y )¢j3/( )33 kd$dy

 

\

 

_q)-1

 

Z/f

28_-y

Ny 1 d2
2/ / :a—Ery)

Z/f

110(d2
2 6y)

116()d2 "

)2 “if“

i=1

wU(
i=1

i=1

—a:z-,(t may )asjonkdzdy )

221-]- (t))¢’1(y)¢j(y)$"d=vdy

 

:1:—xij(t))¢'1vy(y)¢j(y)$kd$dy }

45

 

 

-<I>‘1 § [1 [1 $112 imam — zij(t)>¢a<y>¢;(y>xkdxdy
'=0 0 0 1—1
NV 1 n I I $k x
(120/0 [0 ; 42%me—xij(t>>¢Ny(y)¢J-<y) My /

+<I>-1 [01 [01 f¢1(y)xkdxdy

 

 

 

 

+<I>-1

_(p-l

Ny n

f 22%“)

j=Oi=1

N” n

2 20%“)

j: —Oi=l

N” n

(ZZWJU)

J'=O i=1

 

(NM

ZEWUU)

j:0 i=1

NV n

22%“)

j=0i=1

 

(22%“)

j=0i= 1
Ny n
( 20mm

2 Zea-fit)

j=Oi=1

 

(SEW-(t)

J=Oi=1

46

1(01/0 ka($ij(t)1ya t)¢0(y)¢j(y)dy \

0] ka( a:(.-J-<t )JJJ t>¢1(y)¢J-(y)dy

 

Wfo ka<xJJ( )JJJt)¢Ny(y)¢J-(y)dy )
(of; _qu k—11b2(xJJ-(t)y,yt>¢o(>¢J(y)dy)

flu) file — 111221 (:cJ-J( y t)¢11y>¢J(y)dy

 

40/01ng — 1)b2($ij(t)a y, ”(My (y)¢j(y)dy /

 

-J(if)1/0 84mg), 31’ t) ¢o(y)¢J(y)dy \

 

gym] acoJJ-(t),y’t)JJ(y)¢J(y)dy

8y

 

 

WA admins/:1), y, t) ¢Ny (y)¢j (y)dy /

(4.23)

47

(NZZW'J(thiJ( ”/1C(xlj(t)vth)¢0(y)¢3(y)dy \

J':0 i=1

_qJ—I EEXJJJJW JJJ( n/‘c (JJJ<t),y,t)¢J(y)¢;-(y>dy

J':0 i=1

 

 

KZELMA) JJ(t>/0 c(xJ-J-a),th)¢NJ<y)¢;-(y)dy J

j:0 ’=1

{NZZwJ-JU) ZU)/01;a(d2(x'gg)y’t))¢6(y)¢j(y)dy \

J=Oi= l

 

1 1 3(d2($ij(t) y t))

 

 

 

 

_qJ—I 2:3ng 0 5 6y ’ ’ ¢a(y)¢J(y)dy
:20)“ ”118(d2($ij(t)1yit))¢l ( )¢( )d
K120i=l 'j xikj( 0 2 8y Ny y J y y j

(N J \
Z;wi1(t)$ij(n/0112d ¢0¢de { fol/Olfngyfikdxdy \

"(b—1 szwfllt) SOD/01; d2¢1¢j dy + (1)4 fol [01 f¢1(y)xkdxdy

 

 

 

 

j=0i=1
: 1 1 '
k
\ NZZWJ’J'“ kHz-(t) [01;_d2¢’1vy¢3dy ) \ /0' ‘/0 f¢1vy(y)at dIL'dy )
j:0 {:1

This is a system of Zn x (Ny + 1) ordinary differential equations for the n(Ny + 1)

nodes and n(Ny + 1) weights. Next, we shall pose the initial conditions for solving

 

 

48

equation (4.23). From (4.13), we have
1 1 n

/ Olj(.’13,0)$kd$ =/ zany-(0)6
0 0 i=1

( [O‘JMJJOWJ ) / ng-owntm) \
[01 al(:1:,0):1:kdx ;wi1(0)$n(0)

 

 

 

 

K j: aNy(.’L‘,0)xkdII: ) K :wim(0)$fNy (0) /
By (4.3) and (4.13), we obtain

Ny n

“(513 ,yt) NZZwij(t) “330'“ ))¢J’(y )

j=0i=1

Therefore,

[01 [0' a(mJyJ 0)$k¢z(y)d$dy

lNy n

:fo [0 2821040” $ij(0))¢j(y)$k¢z(y)dxdy
zigfilwn(0)6(x—( xJJ-(O )):c’°ld:z:/0 (My )¢j( ydy)

—Z(§:wta (0))/0 (My )¢J(i‘/

=0 i-l

H.

3

= E: (Z wJJ-(0)fo(0)) ¢(lJJ').

K).
H

(J: — $ij(0))$kd$ = :wJ-jm) k

(4.24)

(4.25)

(4.26)

(4.27)

or,

49

1 1
f0 f0 u(x, y, 0)$k¢z(y)d$dy

/ ZwioWi-com) \

n

Belmont-“1(0)
=(¢(1,0) 40,1) ¢(,,Ny)) i=1

 

 

(:wi~.(0)xfm(0) )

f [01 [01 u(x, y, 0)$k¢0(y)d$dy \
[01 [01 ”(1" y, O)$k¢1(Z/)dxdy

 

 

\fol [Olu($,y20)$k¢1vy(y)d$dy)

/ 440,0) 440,1) 440,414) \ .7.)
_ «((1,0) 40,1) Jaw.) 5141-4025(0)

 

 

 

/ iwmmmom) \

 

( 4W0) ¢<Ny,1) ¢<Ny,Ny) ) \iwmymfifmm) )

{ :wmmnfom) \

$442440)
= (I) i=1

 

 

ngmumfifmm) /

(4.28)

Therefore,

{ :szm) \
:31 wil (0)23?1 (0)

 

 

K :wmy (0)1273,” (0) j

: cp—l

50

( ./01/olu(x’y’0)$k¢0(y)d$dy \
[01 [01 u(x, y, 0)$k¢1(y)d$dy

 

 

K[OI/()1u($,y,0)$k¢1vy(y)dxdyJ

(4.29)

CHAPTER 5

Convergence Results

5.1 Convergence of Semi-Discrete Finite Element

Approximation in y

Let Vh E H6(0, 1) be a finite dimensional subspace as defined in Section 4.1. Given
f E L2(0, 1) and no), 6 Vh, a suitable approximation of the initial datum no 6

L2([0,1]), for each t E [0, T], we attempt to find uh(t) E V), such that

(%,vh) + a(uhwh) = (fHUh) V’Uh ’5 Vin t6 (QT),

(5.1)

uh (O) = U0,»

Lemma 5.1.1 For fixed a: and t, let m, be the linear interpolant of u, that is, 7r,l E Vh,

7r),(yj) = u(yj), 0 Sj g Ny. Then

IIU' - thlo S hHU"||o (5-2)

51

Proof. For any y E (y), yj+1),

11(41)—

h

_1‘/1:1j+1(u:/
— h

= yj+l [311:-
[0 (My) -

IIU' - rill?) =

Ivy—1

= Z /.,’.”“<

J:0 J

= _/:”“{/”’“/yu

|/\

lNy—l

|/\

NV

l/\

J=0
Ny—l

i=0

Uh—(yjH)

52

”(l/j)

 

,3. 2 l.-
231 [3:141

yj+1

= M’Zf

1 UJ+1u(, uh/l/JHWI
= —- dz — .—
y;

'Z())dz

'(w)dwdz
W ))2dy

My) - 7r}.(y))2dy

()dwdz} dy

yJ

,3. f; /f’“ { 5““ ran 124442} { f.. / ......)
yj+1{/y

J

yjH/y lu"( (w )|2dwdz} h2dy

y]

/yJ+1/yj+1 Iu"(w)|2dwdz} dy
31' EU

J

{

|u"(w) Izdw

1
= h2/ |u"(w)|2dw
o

= h"’IIU"II3-

Lemma 5.1.2 Let Jrh be the linear interpolant of u, that is, 7r;l E Vh,7rh(yj) =

u(yj),‘v’j. Then

||u - 7rhllo S hPIIU"|I0- (5-3)

53

Proof. For y E (yj, 31,-1.1), since u(yj) = Jrh(yJ-), we have

31

u(y)-m.(y) = / (u'(z>-vr;.<z>)dz

IU(3/)--7rh(y)l2 = (lift/(Z) -7r§.(Z))d~2)2

y I I 2 y
S In (2) — Jrh(z)| dz/ dz
w 311'

= (y — 313')/yj+1 IU( )- 71:1(z)|2dz

311

1
Ila—mug = / lam-mafia

N,— —1
yj+1| 2
= 23/] )(-7Th y)ldy
3:0 111'
yJ+1( yJ+1 I
s :[y (y— y. )/ Iu’z( #444124sz
h2 NV 1 J 1
= __ 2: [W z)—7r;,(z)|2dz
j=0 yj

= 33/0 lu’(z)—7r§.(z)l2dz

= ‘2—“11' — Willi-

Inequality (5.3) follows at once from (5.2).

Theorem 5.1.1 Ifuo E H6(0,1), f E L2(0, 1) andu satisfies the following condition,

H1433, 30“? S C(IILU(:E, -.t)||3 + IIU($. -,t)l|i), (5-4)
then the energy estimate

max ||u(t )— u,.(t)||g + a [OT ||u(t) — uh(t)ll¥

0<t<T

2 2 2 2 T 2 (5'5)
3 Hue — nan. + 0...}. (Huo,h||1+|luo||1+ l. anodt)

54

holds; where C2m > 0 is a constant independent of h.

Proof. For each t E (0, T] define eh(t) = u(t) — uh(t). Taking v = v), in (3.21), we get
the weak form

(ut, ’Uh) + a(u, vh) = (f, vh) (5.6)

Subtracting (5.1) from (5.6), we obtain

 

(Ea—till) Uh) + “(U — uh’vh) = 0’
(665th) ’ Uh) + a(eh(t)7 Uh) : O (5.7)

For almost any fixed t, choose 1)), = uh(t) — wh, w), E V), in (5.7). For each 6 > O and

for almost any t E [0, T] we find

 

 

 

 

 

, u — w), + w), — uh(t)) + a(eh(t), u — 111,, + w), — uh(t))

,u — 10h) + a(eh(t), ”U. — 112),) + (aegft) , w), — uh) + 0(6),“), 11)), - uh).

 

 

By (5.7)

 

(8eh(t)

at ,wh — Uh) + a(eh(t),wh - Uh) = 0.

55

Therefore, we have

 

   

 

 

 

 

gem). em) + «(2.1021(2)
= :25”, u — wh) + a(eh(t), u — 112),)
_<_ )—( nu — will. + cuehumluum — win.
0
g J; 0 Ha — while + j—Znu — will? + diam”?-

 

 

 

 

Let w). = Jrh(u(t)) be the linear interpolant of u, for t E [0, T]. By Lemma 5.1.1 and

Lemma 5.1.2, we have
IIW) - 7rz.('tt(15))llo S 7h2IIU(t)||2 (5-8)

and

IIU(t) - rh(U(t))lli S 72h2IIU(t)Il§- (5-9)
We thus obtain

1 68h

at 7h2IIUI|2+ —72hQIIU(t)H§+€||eh(t)lli-

0 (5.10)

a «(can elm) + a(ehu), em» 3

 

 

 

 

Using (3.28), we have a(eh(t), eh(t)) 2 oz||eh(t)||¥. If we choose 5 = 55-, then

 

 

1 6
2am ( 164(4)) + )auehumf
1 8(
S ”gt— (6)105) €h(t )+ a(eh(t),eh(t)) (5.11)
06,, 62 oz
<flateh 0:7h2llull +"— 20 72h2llu(t)li + Elle/JUNK-
So
1 d CY 6e
5d—Illeh(t )IIS + Elleh(tlll2< _ a—h 7112))qu + 2—-’y2h2||u(t mg. (5.12)
0

 

 

 

 

56

Integrating over (0, t) for t E (0, T], we obtain

t
2/—IIe1.(II.aIr+§ [0 IIehmIIJdr

ta€:(7' )

c2 T
a, 7h2IIU(T)II2dT+%72h2/O llu(r)||§dr

|/\
o\.o

 

 

 

 

0
2

 

 

 

s .12 [0‘ ( 883:” 0 + Ilu(r)ll§) .1. + 1211225,; [0" llu(r)||§dr
= 'th/OT( 6:35-71 +(1+—)IIU (THIS) (17'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 T aem) 2
< _ 2 2
_ ,0...h [0 (—,,T O+||u(r)||2) .1.
So
1 2 1 2 1 2 T Beh(r) 2
_ .. <_
2lleh(t)llo 2IIe.(o>II3+ 3116.114. 2c..,,1.(/0 37 0+llur()ll2d
Since
lleh(0)||3 = ”no - uh,o||3
and
agha) 2 311(1) _ auha) 2 < 811(t) 2 Buh(t) 2
6t 0 at at 0- 8t 0 6t 0’
it follows that
IIeIt>II2+a/‘IIe(T)II2dr<IIu -.. l|2+C ,sz 8—”2 53—93-114“? dt
’1 0 0 h 1 — 0 h,0 0 0,7 0 at 0 at 0 2 .

 

 

 

 

 

 

 

 

As in (3.31), we have

/.

git—112

 

 

dr<c {IIuo.II¥+ / llflISds}

 

 

0

57

Therefore,

t
wamt+afuavm21

T
s In. — 44.413 + 0...}? f0 (

2

8u(t)
W—

 

 

 

 

t
+ IIu.,.II1+ f0 ||f||3 + ||u(t)||§) J...

0

Using (3.29), (3.31) and (3.32), we obtain

t t
Ilen(t)l|3 + a [0 lleh(r)ll¥dr s lluo — mans + 0.42 {IIUolli + “no.1? + l. llfllfidt)

So

02% ”4(1) — uh(t)||3 + a [T Ilu(t) — um”?

T
s IIuo — mug + 0.. (IIuo,.IIf + lluolli + [0 IIJIISds)

5.2 Convergence of the Gauss-Galerkin Approxi-

mations in x
In Section 4.2, we replaced (II-(2:, t)dx by an n-point dirac-delta function, i.e.,
dyflz. t) = gamma — xI-J(t))drv (5.13)
Now we define moments of the discrete measures by
ml. (t) = [01 xldp?(:z:,t), t E [0,T]. (5.14)
Substituting (5.13) into (5.14), we obtain

m3,n(t) = :wifitflII-(t) (5.15)

 

58

Define

dfl"(:1:, 1) = (d235, t). cit/f(x, t). - - - . duke. t))T

and

midi) : (ml),n(t)1 ml,n(t)1 ' ' ' 2 ley,n(t))T°

For given u(x, y, 0), as in (4.29), we can calculate 752(0) as follows:

/ mI),n(0) \ ( iwiomlxiowl \ f [1]]

i=1
n

 

 

 

 

 

( mi..,..(0) )

We assume that
|a(a:, y, t)| 3 am: + a0

|b2($, y, t)| S b25132 + (1113 + b0

lC($J ya t)l S C0
30(4‘. y. t)
0y
3(d2(x. y. t))
5y

301

 

 

<511

 

 

|d2($, y, t)l S d0-

gw..(2)x£.(0) =qu [0 [0

Igw‘mwlmimm) ) KAI/1.301451261936114 I

(5.15)

(5.17)

nox’qfiodxdy \

noxldldxdy

 

(5.18)

(5.19)

Lemma 5.2.1 Let I be any positive integer. The set {rh£,(t) : n 2 %(l + 1)} is uni-

formly bounded and equicontinuous in t E [0,T] for each I.

59

Proof. We have by (4.23)

53mm N
- 2M p.522 (0)5034] 4001/0 l(1635113 ).y.t)¢s(y)¢J(y)dy)
N1, n
+2Jp,s)22(w.. 2m], -l(l-1)b2(xz-J( > J t)J.(J >J J?/( ) dJ)

3:0 2:1

 

*5: «10.5% in: (005% (t1)/0 6420-53, y’ t)¢s(y)¢J-(y)dy)

j=0i=1

Nu ~ Nv n (5.20)
22252;: (42.124252) / cIJ..<> w w my)

 

Since |¢J(y)l S 1. |¢3(y)l S 1 and

/1/1|f¢s(y) kudde< / / IfIdde (5.21)

—i/ /. 'flzdxdy} {/ / dandy} =||f(t )II.

we have

5,175.5)

Nu ”11 n
<mzzzn.
s=0j=0£=1
Nu NJ n

+pIZZZ wJJ-(t

s=0j=0i=l ((

Nu N14 n

-m:::@.
s=0j=0 i=1
Nu Nu n ((

#1222 wait

s=0j=0i= 1
Nu Nu n

+101 :22 ”hit

s=0j=0 i=1
Nu N1) n

-mzzzb.an

s=0j= 0i=l

+01 :1) f0 lf¢s(y)

< PIZZZ(ij(t)$

s=0j=0i=l
Nu Nu n

-m;zzb.w.

s=0j=01=1
NJ ”11 n

+p1 Z Z: Z (ij (t)$ (t

5=0j=0 i=1
”1 Nu n

+P1 Z Z Z (“a(t

8:0 j=0 i=1
”2 N” n

-mzzzb.m

s=0j=0 :21
NJ Nu n

-m;:;@.m

3:0 j=0 i=1

+I01 ZN; ”f(tlllo

60

:()/0 ll(a1:v.-J( ) + a.)J.(J)J.-(J)IJJ)
f01§1(1_ 1)|(b2:v?.-(t) + 55.5) + b.)J.(J)J.-(J)IJJ)
[(JIIJ. J)J (J)IJJ)
J..( of c.IJ.(;J)J-J< )IJJ)

J51 (t) f 5d1IJ;(J)J.-<J)IJJ)

$Jj(0/11§do|¢.y) J;(J)IJJ)

wk ldxdy

(t)l(a1$,-j(t) + 0.0))
l)—(l — 1)(IJ.J§,.(J) + 515.5(1) + 50))

)Cl)

)xlj (tlco)

1
’(tlzd‘)
1

(”2‘1”)

(5.22)

61

where p1= 0<maxN|(,15(p, s.)| So we have
d I
amme)
(N +1) )lpl Z 21(w‘j(t$ (t)(a1$J'j(t) + 00))
j=0 i=1
1
+2(Ny +1)l 1‘19122(wu M(b$()+b1$ij(t)+bo))
j=0i=1

(N +1) p1c122;(wij(t)x (t))

j:0 i=1

(N +1)p100§::(wij(t)x (t))

J=0i=1
+2(Ny +1)p1d122(wij(t$) 3(0)
J'=0 i=1
+1
(N + 1) d 1
+2( ”’1 °§;(w’ 0) (5.23)
+(NJ +1)101l|f(t)||o
NV
3 (Ny + 1)lp1 Z: (almg’nU) + aomg n1(t))
i=0
1 N"
+§(Ny + 1)l(l — l)p1 Z (bgmgmu) + blmjjnl (t) + bomggflt»
i=0

NV
+(Ny +1)p1(c1+c0+— éd2+ éd1+2d0) )2m§,n(
.7:

HM + 1)p1l|f(t)||o
”1:

N1!
=n12m3,n(t)+n22m§.3(t ()+J32mg-r: t)+n4l|f(t)llo
i=0

3':

= WWW) + 7757731.”1(t)+ 771W J2(t t)+ m||f(t )Ilo,

62
where

n1=(Ny +1)p1 (la1+%l(l—1)b2 + c1+ Co + édg + édl + édo)

772 = (Ny + 1)lp1 (a0 + $0 -— 1)b1)

173 = 5% +1)l(l‘"1)Plbo (524)
7-7? = (UiJ77i,"‘,77J)a i=1,2,3

774 = (Ny +1)/)1-

We define K,- by

 

 

 

 

(7'7?) {774\
Jr

Ki: 77‘ J:1,2,3 *4: 77f (5.25)
K’f/(Nyunuvyn) \"4/(Ny+1)x1

Then we have the following differential inequality

d

a7715.00 s KIT-nut) + mm) + Kama—fit) + IIf(t)IIoJ'J. (5.26)
Define

d .. .. J .. .. _ J

am’fl) = K1m1(t) + K2ml 1(t) + K3771l 2(t) + Ilf(t)||0774

(5.27)

Jug-(0) = m;,.(0), o s J < NJ,

where

63

Defining 11330) = 772’ (t) — 771’ (t) and subtracting (5.27) from (5.26) we obtain

71

53511210) 3 K1 2123.0) + KJMJ-l (t) + Kama-20)
(5.28)
1125(0) = 0, 0 gj 5 N5.
There exists I?"(t) g 0 such that
$193“) = K1 M50) + K2M'f5—1(t) + KJMfiu) + film
(5.29)
125(0) = 0, 0 gj 3 N5.
The solution of (5.29) is
_. .. t _. .. _.
Mm) = eKltM,’5(O) + [0 eK1<t-T>{KJMj,-1(J)+ KJMg-2(J) + 15(7)} dJ ( )
5.30

t .. ... -o
= / (am-T) {mm-1(7) + [gm-2(7) + R'(J)} d7
0

-o t ..
M,15(t) = / emu—T) {R1(T)d7'} S 0
0
—o t -o -o
M§(t) = / eK1<‘-T>{K2M,1(J)+ R2(T)d7'} g 0
0

By induction, Mgn-la) S 0. Therefore,

mm) s fii’(t), l=0,1,2,---,2n—1 (5.31)

64

 

 

 

 

 

 

Define
( 77‘1“) \ (774)
7731—10) 774
M10) 2 5 1'3" = (5.32)
n'i,(t) 7-7.4
40 -o
\m“) )(l+1)(Ny+1)x1 \"4/(z+1)(Ny+1)x1
(K1 K2 K3 0 0 0 0)
0 K1 K2 K3 0 0 0
0 0 K1 K2 0 0 0
A: g g g g g 3 3 . (5.33)
0 0 0 0 K1 K2 K3
0 0 0 0 0 K1K2
(0 0 0 0 0 0 K1)
We have d
d—JM'“) = AM‘0) + llf(t)||oB’
(5.34)
m§(0)=m§,n(0), 0<j<m, 0<k<t
The solution of (5.34) is
.. _. t _.
M'(t) = eA‘M'(0) + [0 eA(‘_T)||f(T)||OBldT. (5.35)
Thus,
fri‘t oo< Wt 00
II ()|| _H ()II (5.36)

.. t _J
S e”A||‘||M’(0)||.. +f e”"”‘“’)||f(T)||o||B’||oodT-
0

65

By (5.25) and (5.32), we have

IIB‘IIJJ = llfi'Jlloo = m. (5.37)
By (5.32) and (5.27) we have
(0 _
||M(O )lloo- — gggglllmm )lloo — gggglllmi. (0 W» (538)

By (5.18) we obtain

—ol 1
llm (0)”... S “(P Maggy,

[01 [01 ”a(ar, y, 0)$’¢J(y)dxdy(

£|I<I>“1||/0 [0 Iu( J,J,0)IJJJJ

5 “(VIII {[01 [01 (1103,21, 0)l2d$dy}% {[01 [01 dzvdyf

= II‘P‘IIHIUoHo-

 

(5.39)

Substituting (5.37), (5.38) and (5.39) into (5.36), we have

“mm”Se'lAIIT”‘I’-1””“°”°+"4if. M” ”W U. ||f( )IIngf

_ 1
=ellAllT||<I> 1||||u0||0+774{2“A” (32llA||t_1)} {f0 ||f( )W}

SeIIAHTII‘I’_1||||uo||o+774{2“2“(BQHAHT—l }{/< |lf( )I|=3dT} cm.)
(5.40)

 

 

where

 

00,15) = e“"“TII<I>"llllluJ||o + J. {5'1” (JZIIAIT —1)}2 {fat ||f('r)||3dr)§ (5.41)

So, for each given l, {77120) : n 2 5(1 + 1)} is uniformly bounded.

Now, we consider the equicontinuity of {172205) : n 2 5(l + 1)} in [0,T]. By the

66

mean-value theorem,

|t2 — t1|, t1,t2 e [0,T], g e (t1,t2). (5.42)

 

d
|m§,n(t2) — 7723,45): = lamb“)

Following the same process leading to (5.26), we also obtain

 

 

 

 

d «I
—- t
.5 ..< > ..
_<. llKlllllTfiMtllloo + llellllTfiL—VtNloo + IlKallllmL—2(t)||oo + llf(t)||o774 (5.43)
s (IIK1||+||K2||+||K3||)C(l,Ny)+ 05,525,. “mum.
Therefore,
lm;,n(t2) — m;,n(t1)l S é”, Nil/)It? _ tlla t1: t2 6 [0) T] (544)

This proves the equicontinuity of {fiz£,(t) : n 2 —;-(l + 1)} in [0, T]. D

Lemma 5.2.2 Given Ny, there exists a sequence kmkn —) 00, and a sequence of

*1

functions {m,(t)} such that for every fixed integer l, we have

lim m;n(t)=mi(t), Vte[0,T]. ‘ (5.45)

[cu—>00

Proof. For each I and j, using the Ascoli-Arzela theorem [17] and Lemma 5.2.1,
we get a subsequence of {mg-Mt) : n _>_ %(l + 1),t E [0,T]} that converges uniformly
to a limit which we denote by m§,,,(t). Taking intersections of these subsequences

successively with respect to j and applying a diagonal selection principle with respect

67

to l, we obtain a sequence kn, such that

lim m;n(t)=m‘(t), Vt€[0,T] a

*
kn—ioo

Given a sequence {mn};',°:0, define the following differences

Aomn :2 mu,
(5.46)
Akmn = A’s-1m“ _ Ak_lmn+1, k =1,2,...

We have the following result concerning with the classical moment problem [10].

Theorem 5.2.1 A necessary and sufi‘lcient condition for the existence of a solution
of the Hausdorff moment problem, i.e., the existence of a unique nonnegative measure
u satisfying

m, 2/01x'du,l= 0,1,2,---,

is that

Akmz 20, k,l=0,1,2,---.

Lemma 5.2.3 Given Ny, for each 0 g j g Ny and for anyt E [0,T], the elements

of the sequence {mg-”(fire

1—0 are the moments of a nonnegative measure, i.e., there

exists a nonnegative measure de,,..(x, t), such that for each I Z 0 and each 0 S j g Ny,

we have

1
/ x‘de,.(x,t)=m§-,(t), Vt€[O,T]. (5.47)
0 ,

68

Proof. For given j and n, {m’- (t),t 2 0} are moments of the measure dug-Kt). By

.7,"

Theorem 5.2.1, the related differences satisfy
0 g AWL“), Vi =0,1,2,---, and lg 2n— 1.
By Lemma 5.2.2, for any I, we have

lim Thin“) = 775’ (t), Vt e [0,T].

a:
kn—mo

Hence

0 g Aimia), vm = 0,1,2,.-..

Thus, for each j, there exists a nonnegative measure de,.(x, t) such that

1
m§-,,,,(t)=/0 x’de,,.(x,t). E]

Let dP.(x,t) = (dP0,,.(x,t),dP1,,.(x,t),---,de,,.(x,t))T. We have the following

corollary.

Corollary 5.2.1 Given Ny, for any f E C[O,1], we have

1

lim f(x)d[i""(x,t)= [01 f(x)dP.(x,t), Vt€[O,T]. (5.48)

kn—ioo 0

Proof. For every fixed integer l 2 0, by Lemma 5.2.2 and Lemma 5.2.3 we have

1 1 _,
lim x’dflk"(x,t) lim m;n(t)=m' (t) = / mans), Vte [0,T]
0

kn—mo o — kn—mo

69

000 is dense in C[O,T].

By the well-known Weierstrass approximation theorem, {x’}

So for any continuous function f,
1 1 _,
Iim f(x)dfl""(x,t) = [0 f(ar)dP.(:r.t), Vt e [0,T] D

kn—yoo 0

Using (5.15) and definition of L'{, we could write (4.22) as follows,

5,55.) = [01 WWW) (5.49)

Lemma 5.2.4 Given Ny, for each I, we have

mun—751(0) = [[01 L'f(x’)dP..(x, s)ds= [Gt/01 L’f(x’)p',.(x,s)dxds. (5.50)

Proof. Integrating (5.49) over (0, t), we get

Similar to the proof of inequality (5.26), we have
1
[o L'i‘(x‘)dfi""(s) s K mil... (t) + Kzn'il:l(t) + mafia) + llf(t)llofi.

By the Lebesgue dominated convergence theorem, Lemma 5.2.2 and Corollary 5.2.1,

we obtain as kn —-> 00,

151(5) —mf,(0) = [0’ A1L§(x‘)dP.(x,s)ds= [at A1L{(xl)fi.(x,s)dx [:1

Theorem 5.2.2 Under all the previous assumptions, given Ny, for any function f E

70

C[0,1], we have
lim f(x)d[I;-‘(x,t) =/01f(x)d'j(x,t)dx (5.51)

Proof. From Lemma 5.2.4, for any l, we have

(xl,p',.(x,t))— x,p,.( x, 0)) )=/0 /01( L*(x )p..( x, s) )dxds

Since {x‘} is dense in G[0,1], we see that {fi.(x,t)} are the weak solution of (4.7).
Since (4.7) has a unique solution, we conclude that the limit of {[Z’cn} is independent
of the choice of subsequences. It follows that the whole sequence is convergent. This

completes the proof of (5.51) . [:1

Theorem 5.2.3 Under all the previous assumptions, for any continuous functions

g(x) and h(y) in [0,1], we have

lim 11m 2] [o h(y)g(x xy)q5j( )duj() ()dy2/0/0u u(,t)xy, ()g)dxdy. (5.52)

N,, —+oo "400

Proof.

23/ [h ((9)993c5a y)d#1)-dy [0 [01:11 Hwy. h(y)g( )dwdy
=2] / h(y)g svy)¢j( )du?(t )-dy 23/0 [Ohm g(x)¢y-()a,-(a:,t)dxdy}
+2]0 [h(y >450; )a.(xt)dxdy— / [u u(,t)xy, Wlhawmdy}

(5.53)

71

For fixed Ny, using Theorem 5.2.2, we get

,,1i_,Ig°{_NZy/01/Olh(y)($)¢j(y)du}‘(t)dy-Z/o [)h(y $)¢j (wag-(113 t)dzdy}

=,,1i_,rg;/01My)¢y(y y){/g(x>( >(du,<)—a.(xt)dx)}dy =0-

Now, we estimate the second term in (5.53). Since

 

x)¢j(y y)aj( x, t) )—dxdy [0 [Cu (x ,ty, )h(y )g(x)dxdy

     

 

 

1 ”V
0 Mg(:)/:y)g h(21/) ({Z¢j(y)aj(x 15) - 14:1: 31. t)}dydx

   

Z ¢j(y)aj($’ t) _ U($, y: t) dydIL‘

 

 

s [019(4){ 011455245? {[01 Eat-(mote t)— amt) dy}2 dz

1:0
1

 

 

2

={ 01|h(y)l2dy}%/olg(x){/01 §E¢j(y)aj(x, t)— u(x,y,t) dy}2dx
{

 

1 % 1
_<_ wordy} { 0 )g(x.-5244:} / [0 z ¢.(y>a.(x. t>— u(x. MI 4.444.
j=0
(5.54)
In Theorem 5.1.1, by choosing 110,), = uo, we get
2 2 2 T 2
Orggym )— 24(4)“. 3 04.4 (lino||1+ [0 Ilfllodt) (555)

We then obtain

2
l T
454430.52], (114.11% [0 115134044
(5.56)

 

 

Z ¢.<y>a.(x, t) — u(x. y, t)

72

Therefore,

Nllirnoo{Z// :1:y)¢>j()aj(x,t)—dxdy fell/Du u(,txy, )hy=()g(x)dxdy} 0

This proves (5.52). C]

 

CHAPTER 6

Numerical Results

In this chapter, we shall consider the following partial differential equations
1 2
ut = —(au)I + §(b u)m + uyy. (6.1)

(6.1) models a family of singular processes with singularities at x = 0 and x = 1. We
shall use the Gauss-Galerkin finite element method to find the solutions. We use the
finite element method in the y-direction with piece-wise linear finite element space
and the Gauss-Galerkin method in the x-direction to solve the proposed problems.
Our numerical results will show that the Gauss-Galerkin finite element method is a

efficient method dealing with a variety of such singular problems.

6.1 Piecewise Linear Finite Element Space

We choose grid points {yj};:_‘_’0 by dividing [0,1] into Ny equal subintervals with h =

1
—,yj = jh,j = 0,1,---,Ny, such that
Ny

0=yo<y1<y2<...<yj<...<yNy=1.

73

74

Let V), C H1(0, 1) be a finite dimensional subspace and ¢j(y) be base functions of Vh

sketched as follows:

 

 

 

 

(bah?!)
l)
1 (
v 3 3 4. Av : : : A, y
0 = yo 311 312 313 yNy—z yNy—1yNy = 1
$10!)
A)
1 4)
: - = c - 4 4 4 A, y
0 == yo yl 3J2 21/3 yNy—Z yNy—1yNy =1
€520!)
T
1 (1

 

L
‘v

V

v 3 3 C A y
0 = yo 291 312 29/3 yNy—z yNy—lyN, =1

 

 

 

0=yo

 

0:310

Elf—1

yj—l

yj—l

yNL—2 yN:_1yNy'_—_ 1

yN:—2 yN;—1y1vy:= 1

76

 

 

 

 

 

6(11)
ll
15
0 - A 1 A A A1
.710 311 92 313 yN,—2 yN,—1 311v,
_10—0
J.
3(y)
4)
11>————o
0 A A A A A A1
.710 311 .712 313 31N,-2 filmy—1 311v,
_10 o———o
l)
¢’2(y)
()
10 0—0
0 A A A A A A1
.710 311 312 313 yNy-z yNy—l yN,
_14) °_____.

 

77

 

 

 

3(y)

4)

10 o———o

0 A A A A A A A A1
310 311 le—l yj yj+1 .71N,,—1 3m,
_10 o—o

v
[Ivy—1(9)

()

1" o——o

0 A A A A A A A A1
310 91 30—1 313‘ UN,—2 yNy—l 311vy

_10

 

78

 

 

My)
4)
10 o———-o
O A A A A A A A A1
310 311 311—1 y] y~,_2 y~;_1 yiv,
...—10
1)
4;" when 0 < y < y
h a _ 1
450(31) = {
01 When yl S 31 S 1
if, when 0 g y < yl
(151(31) = ‘Hfla when 311 S 31 < 292
01 When 312 .<.. 31 S 1
0, when 0 S y < yl
”in, when 3113 31 < 312
(52(31) =
”Egg, when 312 S 31 < 313
01 When 313 S y S 1

(1514(9) =

¢v(y)

¢j+1(31) =

We have

79

when 0 S y < yj_2
when 5... s y < y.-.
when 31j—1 S 31 < 313'
when yj g y S 1
when 0 g y < yj_1
when 31j—1 S 31 < 311'
when 3/j S 31 < 31j+1
when yj+1 S 31 S 1
when 0 g y < y]-
when 313‘ S 31 < yj+1
when yj+1 S 31 < 31j+2

when yj+2 S 31 S 1

when 0 g y < yNy_2

when yN,_2 S y < y~,_1

when y~,_. s y s 1
when 0 S y < er1

when 31Ny—1 S 31 S 1

when0<y<y1

wheny1<y<1

when0<y<y1
wheny1<y<y2

wheny2<y<1

0,
l
h’
hy)= 1
—K’
0,
0,
.1.
h,
0534(9) 1
’H’
0,
0.
l
I h,
(53(9) 1
_H’
0.
0,
l
h’
¢3+1(y)= 1
“H1
0,
0,
(bin—1W): h,
_l
h,
0.
I _—
¢N,(31)- 1
71—1

80

when0<y<y1
wheny1<y<y2
wheny2<y<y3

wheny3<y<1

when 0 < y < yj_2
when yj_2 < y < yj_1
when yj_1 < y < y]-
when y,- < y < 1
when 0 < y < yJ-_1
when yJ-_1 < y < y]-
when y, < y < yj+1
when yj+1 < y < 1
when 0 < y < y,-
when yj < y < y,“
when yj+1 < y < yj+2

when yj+2 < y < 1

when 0 < y < erz
when y~,_2 < y < y~,_1
when erl < y < 1
when 0 < y < er1

when erl < y < 1

81

6.2 A Test Problem

In this section, we shall consider the following partial differential equation:

u, = (x(1 — x)u)“, + uyy (6.2)
with the boundary conditions
1i_rAr(1) xu =1i_rg(1 — x)u = O on (0, T) x (0,1) (6.3)
uy(x,0, t) = uy(x,1,t) = 0 on (0,T) x (0,1) (6.4)
and initial condition
u(x, y, 0) = 1 — cos(7ry) on (2. (6.5)

As we shall see below, the boundary terms corresponding to (4.16) do not drop out.
We shall show how to handle such terms by interpreting them as fluxes across the
boundaries at x = O and x = 1 and the moments of such fluxes. By redefining a
new “total” U including that due to the fluxes and new “total” moments M i, a new
system for such U and M ‘ results. Such a new system can readily be analyzed by the
method in Chapter 5 and convergence results easily follow.

The right hand side of (6.2) is the divergence of the vector field ((x(1 — x)u)x, uy).
Let P(x, y, t) express the flux across x in the positive direction and Q(x, y, t) express

the flux across y in the positive direction at time t respectively. We have

1’(iv. v. t) = -($(1 - IBM). (6-6)

cow) = —u.. (6.7)

82

P(O, y, t),P(1, y, t), Q(x,0, t) and Q(x, 1, t) express the flux across x = 0, x = 1, y = 0
and y = 1 in the positive direction at time t respectively.

By (6.6), we have
P(x, y, t) = -(x(1— x)u)x = —(1 — x)u + xu — x(1— x)ux. (6.8)
So,
P(O,y,t) = —u(0,y,t) and P(1,y,t) = u(1,y,t). (6.9)

Integrating (6.2) over S), integrating by parts and using (6.4) and (6.9) we have

d—t/fou u(,txy, )dxdy =/:/l(x( (1)—xU)m+uyy)dxdy
=/:(IL‘( ()1—33’11. )lez IOdy+/l uyly: de

_/0:_ P(iL‘ yAt) i=0dy (6.10)
=/0 —{P(1,y,t)-P(0,31,t)}dy
: — [01 u(0,y,t)dy —/01v(1,y.t)dy-

Integrating (6.10) over 7 E (0, t), we have

fo1 fo1 “($1311t)d$dy - f51 f51 u(x, y. 0)d$dy
= — f6 fol u(O, y, r)dydr — f; f01u(1,y,r)dydr.

(6.11)

So,

A/OI/lu u(,txy, d)xdy+/0t1/Ou u(0,) ,y, 'r)dydr+A/O/olu (,1y,r r)dydr (6.12)

z]: /1( u (x y,,0)dxdy.
o

83

Multiplying by x and integrating (6.2) over S), integrating by parts and using (6.4)
and (6.9), we have

d 1 1
32/0 [0 u(x,y,t)xdxdy =/0: [01:13“ ()1-23U)U)m$+uyy$)d$d31
1

_/::(:; ()ul—x )lxxlz 0dy+/0 uyl;:0xdx
—- P(xynt )Lwl Ody

1
P(,t)1v, =—/ u(1.y,t)dy.

o

(6.13)

é+w

Integrating (6.13) over r E (0, t), we have

1 l 1 1 t 1
[0/(Au(x.y,t)xdxdy-/O f0 u(x,y,0)xdxdy=—/0 [0 u(1,y,7')dyd7- (6.14)

So,

1 1 t 1 1 1
f0A/Ou($1y,t)xdxdy+/O/O11(1131,T)dydr=/ / u(x,y,0)xdxdy (6.15)
o 0

where [Gt/01 u(0,y,'r)dydr and fat/Olu(1,y,r)dydr represent the moments of the
fluxes across the boundaries x = 0 and x = 1.

Multiplying by x", k = 2, 3, - - - , 2n — 1, integrating (6.2) over S), integrating by parts
and using (6.4) and (6.9), we have

d—dt/(1:/() u(x, y, t)x"dxdy f0 1/()((."10(1 -310)’U1)’U1)mfll?’c +Uyy$kld$d31
=/0117((17()311-$ 30x17]: xzody— [015d (1—$)Uk$k 1|x= Ody
+/011/01x( (1 — x)uk(k — 1)x" :dxldy +A/0 uylyzoflldx

—- —/ -—P<x 4.545114% [0 [0 x(1 ~x)uk(k — 5121-2114411
_/_ p(1 yflflfldy+/AA:/:x((1-x)uk(k—1)x k‘zdxdy

_ —A/01u(1 y,t)dy+/01/lx(1(——x)uk(k1)xk‘2dxdy.

(6.16)

 

84

Integrating (6.16) over r E (0, t), we have

[All/Du u(,txy,) )xkdxdy fofou u(,Oxy, 0)xkdxdy

= _fo [01 u (1,3),7 )dydr+/0 [01 A/01( x (1 -—x)u(x,y,r)k(k —1)xk_2dxdydr.
(6.17)

So,

fol/u u(,t)xy, xkdxdy+/O/0:u (1,y, )dydr

2/09 [()u( ux y,,O )xkdxdy+/O [0 [0x( (l—x) (x ,ry, r)k(k— 1)xk—2dxdydr
(6.18)

On the other hand, we can rewrite (6.10), 6.13) and (6.16) as

i [0‘ [01 {141,115+ [0‘ am, r)6(4)dr + [0‘ u(x, y, r)6(:v —1)dr}dxdy = 0
(6.19)

g, [01 fol {1134.11.15 + f0‘u(x,y.1)6(4)dr + [0‘ u(x, y, we: — 1W} xdmdy = 0
(6.20)

and, fork=2,3,~-,2n—1,

l 1 t t
Slit/o [o {u (:12, 31,15 )+/0u( ux ,ry, T)(5(x ()dT+/ ”(3313/17)5(33- 1)d7'} xkdxdy

=/01/o x,(1—x){u(xy,t)+/OU u(.xy,r W)5()d7’

+/0‘u(.,y,.)5(._1)d.}k(k_1)..k 2.1.5.).
(6.21)

Integrating (6.19), (6.20) and (6.21) over r E (0,t), we have

A1A1{U($,y,t)+/1)tu(x, y, r)6(x)dr + [0‘ u(x, y, r)6(x — 1)dr} dxdy (6 22)
=/01/01{u(x,31,0)+/00 1417,31, r)6(x)dr + [00 u(x, y, r)6(x —- 1)dr} dxdy -

85

[01 01{u(x,y,t)+/otu(g;,y,r)6(x)dr + [otu($,y,r)6(x —1)dr}xdxdy

= [01 [)1 {u(x, y, 0) + f00u(x, y, r)6(x)dr + A/00u(x, y, r)6(x —1)dr}xdxdy
(6.23)

and, fork=2,3,-~,2n—1,

[IA/ol{u(:v,31,t) +A/Otu(x,y,r)6(x)d»r + [()tu($ay,’r)6($ _ 1)dr} xkdxdy

0

=A/01/01{u(x,y,O)+/00u(x,y,r)6(x)dr + [00 u(x, y, r)6(x —1)dr}xkdxdy

[OI/011:“1 — x) {u(x, y,r) ‘1']; “(33, y, T')6(x)dr'
+ /07 u(x, y, TIM” " 1)d7"}k(k - 1)xk'2drdxdy.

+

(6.24)

We now let
t t
U(x, y, t) = u(x, y,t) +/ u(x, y, r)6(x)dr +/ u(x,y, r)6(x — 1)dr (6.25)
0 0
We can the rewrite (6.19), (6.20) and (6.21) as
d 1 1
— f / U(x,y,t)dxdy = 0 (6.26)
dt 0 o

1 1
if / U(x, y, t)xdxdy = 0 (6.27)
dt 0 o

and, fork=2,3,---,2n—1,

d 1 1 k 1 1 k4
— = — — 6.28
dt [0 [0 WWW dwdy f0 [0 x(1 x)U(:v,y.t)k(k 1):.- dxdy ( )

We can also rewrite (6.22), (6.23) and (6.24) as
1 1 1 1
f / U(x,y,t)dxdy = / / U(x,y,0)dxdy (6.29)
o o o 0

[01 [01 U(x, y, t)xdxdy = /01 [01 U(x, y, 0)xdxdy (6.30)

86

and, fork=2,3,---,2n—1,

[OI/1U(x, y, tx) kdxdyzA/Ol/OIU (x ,xy,0) kdxdy

+/1/01/0tx(1— 33) U(a: y”;- T)k(k _ 1)xk— 251761113613}. (6.31)

Let us now calculate a few quantities:

(1) The exact solution of (6.2) with boundary conditions (6.3),(6.4) and initial

condition (6.5) is

 

u(x,y,t) = e'2‘ — cos(7ry)e‘("2+2)t on (0,T) x Q (6.32)
(2)
[If Okdd— 1 15—012 2 1 (633)
OOU($,y,)$$y—k+1, —'911 7” ' '
(3)
1 1
/ u(0.11.1011: / U(1.y.t)dy=e‘2‘. (6134)
o o
(4)
1 1 t 1 1
f/u(0,y,r)dydr=/ / u(1,y,r)dydr=—(1—e‘2‘). (6.35)
o o o o 2
(5)
1 1 k 1 —2t
f/u(x,y,t)x dxdy= e , k=0,1,2,---,2n—1. (6.36)
0 o k+1

We denote by M i(t) the ith total moment

1 l .
= / / U(x,y,t)x'dxdy. (6.37)
0 0

Then we have

87

(6)

11 11 1
U ”Okdd=//' ,pkad=—__y_ .H _,
(1,/0 ($111171?! Commit/M7311 k+1, 0,1, ,2n1
@3&

AM) 1/1 U< )4 d
t 2/ x, ,t x 1
1 1 0 0 y t J t
:/0 f0 {u(x,y,t)-+1)u(x,y,r)6(x)dr+/0 u(x,y,r)6(x—1)dr}dxdy
1 1 t 1 t 1
= 0 f0 u(x,y,t)dxdy+/0 l0 u(0,y,r)dydr+/ ./ “(1,311Tld31d7'
0 0

(1— (3—2t) + —(1 — e—Qt) =1
wsm

M1“) =/01/01 U($,y,t)xdxdy
=[01/0:{u( (x ,y,) t)+A/0tu( u(x ,ry, r))6(x dT+/O‘u ($,y,r)6(x—1)dr}xdxdy

:/1 A/0‘1u11,($ y’t)1$d$dy+/ot [01 u(l y,7' 7')dyd7'

H6—2t+ 1(--1 e‘2‘)=

=2 2 2

(6.40)

(9) FOIk=2,---,2n—1,

Mk(1)=/1/1U($y,tx) kdxdy

=A/0:/0:U({u,t)xy, 117/011145133117 )(5x()dr+/0tu x(,y,r)6(x—1)dr}xkdxdy
:follfolm Mxy’ )zkd‘vdy‘tflflo( (131,7 )Sydr

1

—2t —2t
=k+le +2(1 8 1
(6.41)

For the steady state solution, we have

88

(10)
1 1
f0 u(o,y,oo)dy= f0 u(1,y,oo)dy=o (6.42)
(11)
oo 1 oo 1 1
/ /u(0,y,r)dydr=/ [u(1,y,r)dydr=— (6.43)
0 0 0 0 2
(12)
1 1
f0 f0 u(x,y,oo)xkdxdy=0, k=0,1,2,---,2n—1 (6.44)
This implies that
U((L‘,y, 00) : 01 (23,31) 6 Q (6'45)

In this test problem, x = 0 and x = 1 are exit boundaries. For the steady state, u
will “pile” at x = 0 and x = 1 with a Dirac-delta function singularity formed at x = 0
and x = 1.

(13)

M°(oo) =1 and Mk(oo) = %,

k=1,2,---,2n—l. (6.46)
We apply the Gauss-Galerkin Finite Element method using finite element approx—
imations in the y-direction and Gauss-Galerkin approximations in the x-direction.
We divide y E [0, 1] into four equal subintervals. Then we have five grid lines
yj = jh, j = 0,1,2,3,4,h = 0.25. For simplicity we use the Neumann boundary
conditions to eliminate the unknowns along the grid lines y = 0 and y = 1 (though
this is not necessary as the Neumann conditions are “natural” .) In the x-direction we
use both three nodes labeled by *, + and x and five nodes labeled by <>, 0, x, + and
*, respectively, for each grid line y = yj. We only show the numerical results along
grid line y = 0.5. The results along grid lines y = 0.25 and y = 0.75 are similar. We

use =11, + and x (or O, o, x, + and =11) to indicate the nodes moving to the boundary

x = 0 , interior point(s) and the boundary x = 1.

89

We difine by m3-An(t) the ith “moment” along the jth grid line y = y,,

mgAn(t) = [01 x'duflx, t) (6.47)

and by M},(t) the ith “total moment”

"v 1 1 ,
M:.(1)=;/0 / m'¢.(1)d11;~‘(x,1)dy (6.48)

Figure 6.1 shows the movement of the three nodes as t increases. Table 6.1 shows the
changes of the nodes *, + and x as t increases. Table 6.2 shows the changes of the
weights at the three nodes *, + and x as t increases. Table 6.3 shows the changes of
the ith “moment” m§,(y, t) at y = 0.5 as t increases. Table 6.4 shows the changes of
the “total moment ” M,’,(t) as t increases.

Figure 6.2 shows the movement of the five nodes 0, o, x, + and =1: as t increases.
Table 6.5 shows the changes of the five nodes 0, o, x, + and :1: as t increases. Table
6.6 shows the changes of the weights at the five nodes 0, o, x, + and * as t increases.
Table 6.7 shows the changes of the ith “moment” m§,(y, t) at y = 0.5 as t increases.
Table 6.8 shows the changes of the “total moment ” Mf,(t) as t increases. Table 6.9
shows the changes of the exact total moment M 1'(t) as t increases. Table 6.10 shows
the changes of the errors between the total moment M 1(t) and their approximation
Mn(t) as t increases. Table 6.11 shows the changes of the relative errors between the
total moment M 1(t) and their approximation Mn(t) as t increases. We observe the

following results:

(1) Table 6.8 shows that M2(t) = 1 and M; (t) = 0.5 for t 2 0. This shows that the

approximation of the 0th and lst total moments equal the exact total moments.
(2) At t = 1.5, the solution reaches the steady state based on the tolerance chosen.

(3) The solution becomes uniform in y as t increases.

90

(4) The solution approaches zero in the interior and “piles” up at the boundary

x = 0 and x = 1 as t increases.
(5) The weights for the interior nodes tend to zero as t increases.

(6) The weights at the other two nodes tend to 0.5 as t increases. They are the

amounts of the fluxes leaking out from x = 0 and x = 1.

(7) Tables 6.10 and 6.11 show that the errors and relative errors between the exact
total moments and their approximations are very small. So the approximation

is very accurate.

(8) Table 6.4 and 6.8 shows that there is no significant difference for the approxi-
mations of the total moments between using three nodes and using five nodes

in the x direction.

91

Table 6.1. Gauss-Galerkin Finite Element Method: Changes of the three nodes 1:, +
and x at y = 0.5 as t increases

 

time

131

$2

$3

 

0.00

0.1127016654

0.5

0.8872983346

 

0.05

0.08265386728

0.5

0.9173461327

 

0.10

0.06390525119

0.5

09360947488

 

0.15

005111110621

0.5

09488888938

 

0.20

004185349059

0.5

09581465094

 

0.25

003487139567

0.5

09651286043

 

0.30

002944063013

0.5

09705593699

 

0.35

002511492851

0.5

09748850715

 

0.40

002160400788

0.5

09783959921

 

0.45

00187107586

0.5

09812892414

 

0.50

001629656768

0.5

09837034323

 

0.55

001426107726

0.5

09857389227

 

0.60

0.01252981511

0.5

09874701849

 

0.65

001104633973

0.5

09889536603

 

0.70

0009767086224

0.5

09902329138

 

0.75

0008657887541

0.5

09913421125

 

0.80

0007691567891

0.5

09923084321

 

0.85

0006846241354

0.5

09931537586

 

0.90

0006104085625

0.5

09938959144

 

0.95

0005450442828

0.5

09945495572

 

1.00

0004873149834

0.5

09951268502

 

1.05

0004362032427

0.5

09956379676

 

1.10

0003908518293

0.5

09960914817

 

1.15

0003505337425

0.5

09964946626

 

1.20

0003146287663

0.5

09968537123

 

1.25

0002826049383

0.5

09971739506

 

1.30

0002540037654

0.5

09974599623

 

1.35

0002284283277

0.5

09977157167

 

1.40

0002055336282

0.5

09979446637

 

1.45

0001850187067

0.5

09981498129

 

 

1.50

 

 

0001666201483

 

0.5

 

09983337985

 

 

92

Table 6.2. Gauss-Galerkin Finite Element Method: Changes of the weights at the
three nodes 11:, + and x at y = 0.5 as t increases

 

time

011

£02

013

 

0.00

02777777778

04444444444

02777777778

 

0.05

02851871392

04296257215

0.2851871392

 

0.10

02992481303

04015037395

0.2992481303

 

0.15

03148991793

03702016414

0.3148991793

 

0.20

03304744604

03390510791

0.3304744604

 

0.25

03453345547

03093308906

03453345547

 

0.30

03592303075

0281539385

0.3592303075

 

0.35

03720822962

02558354075

0.3720822962

 

0.40

03838908217

02322183565

0.3838908217

 

0.45

03946948176

02106103648

0.3946948176

 

0.50

04045515536

0.1908968929

0.4045515536

 

0.55

04135260497

0.1729479006

0.4135260497

 

0.60

04216853657

0.1566292687

0.4216853657

 

0.65

04290954774

0.1418090453

0.4290954774

 

0.70

04358195877

0.1283608245

0.4358195877

 

0.75

04419172578

01161654845

0.4419172578

 

0.80

0.4474440172

01051119655

04474440172

 

0.85

04524512591

009509748189

0.4524512591

 

0.90

04569863005

008602739892

0.4569863005

 

0.95

04610925419

007781491628

0.4610925419

 

1.00

 

04648096777

007038064451

0.4648096777

 

1.05

04681739356

006365212877

0.4681739356

 

1.10

04712183232

005756335362

0.4712183232

 

1.15

04739728751

005205424985

0.4739728751

 

1.20

04764648919

004707021628

0.4764648919

 

1.25

04787191679

00425616641

0.4787191679

 

1.30

0480758206

00384835879

0.480758206

 

1.35

 

04826024173

003479516532

0.4826024173

 

1.40

04842703071

00314593858

0.4842703071

 

1.45

0485778646

002844270799

0.485778646

 

 

1.50

 

04871426277

 

00257147447

 

0.4871426277

 

 

93
6.3 Dependence of Solution upon Parameters in

the Singularities

In this section, we shall consider the following partial differential equations:
ut = (xp(1 — x)qu)m + uyy. (6.49)

The right hand side of (6.49) is the divergence of the vector field ((x”(1 — x)qu),,, uy).
Again, let P(x, y, t) express the flux across x in the positive direction and Q(x, y, t)

express the flux across y in the positive direction at time t respectively, we have
P($?y1t) : _($p(1_ $)qu)x (6'50)

00.4.1) = -.., (6.51)

P(O, y, t), P(1,y, t), Q(x,0, t) and Q(x, 1, t) express the fluxes across x = 0, x = 1, y =
0 and y = 1 in the positive direction at time t respectively.
Model I We consider the following initial-boundary problem with different parame-
ters p and q:

u, = (xp(1 — x)"u)m + uyy in (0,T) x (2 (6.52)

with the boundary conditions
(40.11.13) = U(1.y.t) = 0 0n (0. T) X (0.1). (6-53)

uy(x,0,t) = uy(x,1,t) = 0 on (0,T) x (0,1), (6.54)

94

Table 6.3. Gauss-Galerkin Finite Element Method: Changes of the “moment”
m;(y, t) with three nodes *, + and x at y = 0.5 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time mo m1 m2 m3 m4 m5
0.00 1 0.5 0.333333 0.25 0.2 0.166667
0.05 l 0.5 0.349347 0.27402 0.228824 0.198693
0.10 1 0.5 0.363821 0.295732 0.254878 0.227642
0.15 1 0.5 0.376905 0.315358 0.278429 0.25381
0.20 1 0.5 0.388732 0.333098 0.299718 0.277464
0.25 1 0.5 0.399423 0.349134 0.318961 0.298845
0.30 1 0.5 0.409086 0.363629 0.336355 0.318172
0.35 l 0.5 0.417821 0.376731 0.352078 0.335642
0.40 1 0.5 0.425717 0.388575 0.36629 0.351433
0.45 1 0.5 0.432854 0.399281 0.379137 0.365707
0.50 1 0.5 0.439305 0.408958 0.390749 0.37861
0.55 1 0.5 0.445137 0.417705 0.401246 0.390273
0.60 1 0.5 0.450408 0.425612 0.410734 0.400816
0.65 1 0.5 0.455173 0.432759 0.419311 0.410345
0.70 1 0.5 0.45948 0.439219 0.427063 0.418959
0.75 1 0.5 0.463373 0.445059 0.434071 0.426745
0.80 1 0.5 0.466892 0.450338 0.440405 0.433784
0.85 1 0.5 0.470073 0.455109 0.446131 0.440146
0.90 1 0.5 0.472948 0.459422 0.451307 0.445896
0.95 1 0.5 0.475547 0.463321 0.455985 0.451095
1.00 1 0.5 0.477897 0.466845 0.460214 0.455793
1.05 1 0.5 0.48002 0.470031 0.464037 0.460041
1.10 1 0.5 0.48194 0.47291 0.467492 0.46388
1.15 1 0.5 0.483675 0.475513 0.470615 0.46735
1.20 1 0.5 0.485244 0.477866 0.473439 0.470487
1.25 1 0.5 0.486661 0.479992 0.475991 0.473323
1.30 1 0.5 0.487943 0.481915 0.478297 0.475886
1.35 1 0.5 0.489101 0.483652 0.480383 0.478203
1.40 l 0.5 0.490149 0.485223 0.482267 0.480297
1.45 1 0.5 0.491095 0.486643 0.483971 0.48219
1.50 1 0.5 0.491951 0.487926 0.485511 0.483901

 

 

 

 

 

 

 

 

 

95

Table 6.4. Gauss-Galerkin Finite Element Method: Changes of the “total moment”
Mi,(t) with three nodes *, + and x at as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5
0.00 1 0.5 0.333333 0.25 0.2 0.166667
0.05 1 0.5 0.349347 0.27402 0.228824 0.198693
0.10 1 0.5 0.363821 0.295732 0.254878 0.227642
0.15 1 0.5 0.376905 0.315358 0.278429 0.25381
0.20 1 0.5 0.388732 0.333098 0.299718 0.277464
0.25 1 0.5 0.399423 0.349134 0.318961 0.298845
0.30 1 0.5 0.409086 0.363629 0.336355 0.318172
0.35 1 0.5 0.417821 0.376731 0.352078 0.335642
0.40 1 0.5 0.425717 0.388575 0.36629 0.351433
0.45 l 0.5 0.432854 0.399281 0.379137 0.365707
0.50 1 0.5 0.439305 0.408958 0.390749 0.37861
0.55 1 0.5 0.445137 0.417705 0.401246 0.390273
0.60 1 0.5 0.450408 0.425612 0.410734 0.400816
0.65 1 0.5 0.455173 0.432759 0.419311 0.410345
0.70 1 0.5 0.45948 0.439219 0.427063 0.418959
0.75 1 0.5 0.463373 0.445059 0.434071 0.426745
0.80 1 0.5 0.466892 0.450338 0.440405 0.433784
0.85 1 0.5 0.470073 0.455109 0.446131 0.440146
0.90 1 0.5 0.472948 0.459422 0.451307 0.445896
0.95 1 0.5 0.475547 0.463321 0.455985 0.451095
1.00 1 0.5 0.477897 0.466845 0.460214 0.455793
1.05 1 0.5 0.48002 0.470031 0.464037 0.460041
1.10 1 0.5 0.48194 0.47291 0.467492 0.46388
1.15 1 0.5 0.483675 0.475513 0.470615 0.46735
1.20 l 0.5 0.485244 0.477866 0.473439 0.470487
1.25 1 0.5 0.486661 0.479992 0.475991 0.473323
1.30 1 0.5 0.487943 0.481915 0.478297 0.475886
1.35 1 0.5 0.489101 0.483652 0.480383 0.478203
1.40 1 0.5 0.490149 0.485223 0.482267 0.480297
1.45 1 0.5 0.491095 0.486643 0.483971 0.48219
1.50 1 0.5 0.491951 0.487926 0.485511 0.483901

 

 

 

 

 

 

 

 

 

96

Table 6.5. Gauss-Galerkin Finite Element Method: Changes of the five nodes 0, o,
x, + and a: at y = 0.5 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time .761 $2 $3 1134 $5

0.00 0.046910077 0.2307653449 0.5 0.7692346551 0.953089923
0.05 0.023013041 0.2191331593 0.5 0.7808668407 0.976986958
0.10 0.014685321 0.2160106921 0.5 07839893079 0985314678
0.15 0010483349 02145721597 0.5 07854278403 0989516650
0.20 0992038063 07862514012 0.5 02137485988 00079619366
0.25 0006288393 02132174718 0.5 07867825282 0.993711606
0.30 0005101903 02128481441 0.5 07871518559 0994898096
0.35 0004220822 02125776662 0.5 07874223338 0995779177
0.40 0996456245 07876280313 0.5 02123719687 0003543754
0.45 0003009645 02122110044 0.5 07877889957 0996990355
0.50 0997420468 07879177936 0.5 02120822064 0002579531
0.55 0997772625 07880227035 0.5 0.21 19772965 0002227374
0.60 0998064888 07881094017 0.5 02118905983 0001935111
0.65 0001689814 02118180894 0.5 07881819106 0998310185
0.70 0998518012 0788243161 0.5 0211756839 0001481988
0.75 0998695503 07882953389 0.5 0.2117046612 0001304496
0.80 0001151877 0211659892 0.5 0788340108 0998848122
0.85 0998980124 07883787575 0.5 02116212425 0001019875
0.90 0000905124 02115876985 0.5 07884123015 0999094875
0.95 0000804931 02115584512 0.5 07884415489 0999195068
1.00 0000717112 02115328475 0.5 07884671526 0999282887
1.05 0999360118 07884896454 0.5 0.2115103546 0000639881
1.10 0999428241 07885094665 0.5 0.2114905335 0000571758
1.15 0000511513 02114730193 0.5 07885269808 09994884863
1.20 0999541887 07885424939 0.5 0.2114575061 0000458112
1.25 0999589318 07885562638 0.5 0.2114437362 0000410681
1.30 0999631524 07885685095 0.5 0.2114314905 0000368475
1.35 0999669141 0788579418 0.5 0211420582 0000330858
1.40 0000297284 02114108501 0.5 07885891499 0999702715
1.45 0000267278 02114021563 0.5 07885978436 0999732721
1.50 0000240431 02113943805 04999999998 07886056192 0999759568

 

 

97

Table 6.6. Gauss-Galerkin Finite Element Method: Changes of the weights at nodes
0, o, x, + and * at y = 0.5 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time (.421 (1)2 (.03 L04 LL15

0.00 0118463442 0239314335 0284444444 0.239314335 0.1 1846344
0.05 0139977042 0.225973313 0268099287 0225973313 013997704
0.10 0170874697 0206626649 0244997305 0.206626649 017087469
0.15 0200965839 0.187750113 0222568094 0187750113 020096583
0.20 0228907253 0170212913 0201759666 0.170212913 0.228907253
0.25 0254494478 0154150051 0182710939 0.154150051 0.254494478
0.30 0277794365 0139521743 0165367781 0.139521743 0.277794365
0.35 0298953252 0126236970 0149619555 0.126236970 0.298953252
0.40 0318138876 0114190794 0135340657 0.114190794 0.318138876
0.45 0335519514 0103277744 0122405482 0103277744 0.335519514
0.50 0351255856 0093397020 0110694245 00933970208 0.351255856
0.55 0365497910 0084454487 0100095204 0.084454487 0.365497910
0.60 0378384055 0076363286 0090505316 0.076363286 0.378384055
0.65 0390041077 0069043827 0081830190 0.069043827 0.390041077
0.70 0400584668 0062423478 0073983707 0.062423478 0.400584668
0.75 0410120113 0056436136 0066887501 0.056436136 0.410120113
0.80 0418743061 0051021748 0060470380 0.051021748 0.418743061
0.85 0426540302 0046125821 0054667753 0.046125821 0.426540302
0.90 0433590516 0041698952 0049421061 0.041698952 0.433590516
0.95 0439964991 0037696382 0044677252 0.037696382 0.439964991
1.00 0445728284 0034077575 0040388279 0.034077 57 5 0.445728284
1.05 0450938838 0030805835 0036510650 0.030805835 0.450938838
1.10 0455649546 0027847952 0033005002 0.027847952 0.455649546
1.15 0459908267 0025173873 0029835718 0.025173873 0.459908267
1.20 0463758298 0022756413 0026970575 0.022756413 0.463758298
1.25 04672388082 0020570978 0024380426 0.020570978 0467238808
1.30 047038522 0018595325 0022038909 0.018595325 0.47038522
1.35 0473229572 0016809337 0019922181 0.016809337 0.473229572
1.40 0475800836 0015194822 001800868 0.015194822 0.475800836
1.45 0478125214 0013735330 0016278911 0.013735330 0.478125214
1.50 0480226393 0012415985 0014715243 0.012415985 0.480226393

 

 

 

 

 

 

 

 

98

Table 6.7. Gauss-Galerkin Finite Element Method: Changes of the “moment”
m:,(y, t) with five nodes 0, o, x, + and * at y = 0.5 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time m0 m1 m2 m3 m4 m5 m6 m7 m8 mg
0.00 1 0.5 0.333 0.25 0.2 0.1667 0.1429 0.125 0.111 0.1
0.05 1 0.5 0.349 0.274 0.228 0.1987 0.1772 0.161 0.148 0.138
0.10 1 0.5 0.363 0.295 0.254 0.2276 0.2082 0.193 0.182 0.173
0.15 1 0.5 0.376 0.315 0.278 0.2538 0.2362 0.223 0.212 0.204
0.20 1 0.5 0.388 0.333 0.299 0.2775 0.2616 0.249 0.240 0.233
0.25 1 0.5 0.399 0.349 0.319 0.2988 0.2845 0.273 0.265 0.258
0.30 1 0.5 0.409 0.363 0.336 0.3182 0.3052 0.295 0.287 0.281
0.35 1 0.5 0.417 0.376 0.352 0.3356 0.3239 0.315 0.308 0.302
0.40 1 0.5 0.425 0.388 0.366 0.3514 0.3408 0.332 0.326 0.321
0.45 1 0.5 0.432 0.399 0.379 0.3657 0.3561 0.348 0.343 0.338
0.50 1 0.5 0.439 0.409 0.390 0.3786 0.3699 0.363 0.358 0.354
0.55 1 0.5 0.445 0.417 0.401 0.3903 0.3824 0.376 0.372 0.368
0.60 1 0.5 0.450 0.425 0.410 0.4008 0.3937 0.388 0.384 0.381
0.65 1 0.5 0.455 0.432 0.419 0.4103 0.4039 0.399 0.395 0.392
0.70 1 0.5 0.459 0.439 0.427 0.419 0.4132 0.408 0.405 0.4028
0.75 1 0.5 0.463 0.445 0.434 0.4267 0.4215 0.417 0.414 0.412
0.80 1 0.5 0.466 0.450 0.440 0.4338 0.4291 0.425 0.422 0.420
0.85 1 0.5 0.470 0.455 0.446 0.4401 0.4359 0.432 0.430 0.428
0.90 1 0.5 0.472 0.459 0.451 0.4459 0.442 0.439 0.436 0.435
0.95 1 0.5 0.475 0.463 0.456 0.4511 0.4476 0.445 0.442 0.441
1.00 1 0.5 0.477 0.466 0.460 0.4558 0.4526 0.450 0.448 0.447
1.05 1 0.5 0.48 0.47 0.464 0.46 0.4572 0.455 0.453 0.452
1.10 1 0.5 0.481 0.472 0.467 0.4639 0.4613 0.459 0.457 0.456
1.15 l 0.5 0.483 0.475 0.470 0.4674 0.465 0.463 0.461 0.460
1.20 1 0.5 0.485 0.477 0.473 0.4705 0.4684 0.466 0.465 0.464
1.25 1 0.5 0.486 0.48 0.476 0.4733 0.4714 0.47 0.468 0.468
1.30 1 0.5 0.487 0.481 0.478 0.4759 0.4742 0.472 0.471 0.471
1.35 1 0.5 0.489 0.483 0.480 0.4782 0.4766 0.475 0.474 0.473
1.40 1 0.5 0.490 0.485 0.482 0.4803 0.4789 0.477 0.477 0.476
1.45 1 0.5 0.491 0.486 0.484 0.4822 0.4809 0.48 0.479 0.478
1.50 1 0.5 0.492 0.487 0.485 0.4839 0.4828 0.481 0.481 0.480

 

 

99

Table 6.8. Gauss-Galerkin Finite Element Method: Changes of the “total moment”
M;(t) with five nodes 0, o, x, + and a: as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5 M6 M7 M8 M9
0.00 1 0.5 0.333 0.25 0.2 0.1667 0.1429 0.125 0.1111 0.1
0.05 1 0.5 0.349 0.274 0.228 0.1987 0.1772 0.161 0.1485 0.1384
0.10 1 0.5 0.363 0.295 0.254 0.2276 0.2082 0.1936 0.1822 0.1732
0.15 1 0.5 0.376 0.315 0.278 0.2538 0.2362 0.223 0.2128 0.2046
0.20 1 0.5 0.388 0.333 0.299 0.2775 0.2616 0.2496 0.2404 0.233
0.25 1 0.5 0.399 0.349 0.319 0.2988 0.2845 0.2737 0.2653 0.2586
0.30 l 0.5 0.409 0.363 0.336 0.3182 0.3052 0.2954 0.2879 0.2818
0.35 1 0.5 0.417 0.376 0.352 0.3356 0.3239 0.3151 0.3082 0.3028
0.40 1 0.5 0.425 0.388 0.366 0.3514 0.3408 0.3329 0.3267 0.3217
0.45 1 0.5 0.432 0.399 0.379 0.3657 0.3561 0.3489 0.3433 0.3388
0.50 1 0.5 0.439 0.409 0.390 0.3786 0.3699 0.3634 0.3584 0.3543
0.55 1 0.5 0.445 0.417 0.401 0.3903 0.3824 0.3766 0.372 0.3683
0.60 1 0.5 0.450 0.425 0.410 0.4008 0.3937 0.3884 0.3843 0.381
0.65 1 0.5 0.455 0.432 0.419 0.4103 0.4039 0.3991 0.3954 0.3924
0.70 1 0.5 0.459 0.439 0.427 0.419 0.4132 0.4088 0.4055 0.4028
0.75 1 0.5 0.463 0.445 0.434 0.4267 0.4215 0.4176 0.4145 0.4121
0.80 1 0.5 0.466 0.450 0.440 0.4338 0.4291 0.4255 0.4227 0.4205
0.85 1 0.5 0.470 0.455 0.446 0.4401 0.4359 0.4327 0.4302 0.4282
0.90 1 0.5 0.472 0.459 0.451 0.4459 0.442 0.4391 0.4369 0.4351
0.95 1 0.5 0.475 0.463 0.456 0.4511 0.4476 0.445 0.4429 0.4413
1.00 1 0.5 0.477 0.466 0.460 0.4558 0.4526 0.4503 0.4484 0.447
1.05 1 0.5 0.48 0.47 0.464 0.46 0.4572 0.455 0.4534 0.452
1.10 1 0.5 0.481 0.472 0.467 0.4639 0.4613 0.4594 0.4579 0.4567
1.15 1 0.5 0.483 0.475 0.470 0.4674 0.465 0.4633 0.4619 0.4608
1.20 1 0.5 0.485 0.477 0.473 0.4705 0.4684 0.4668 0.4656 0.4646
1.25 1 0.5 0.486 0.48 0.476 0.4733 0.4714 0.47 0.4689 0.468
1.30 1 0.5 0.487 0.481 0.478 0.4759 0.4742 0.4729 0.4719 0.4711
1.35 1 0.5 0.489 0.483 0.480 0.4782 0.4766 0.4755 0.4746 0.4738
1.40 1 0.5 0.490 0.485 0.482 0.4803 0.4789 0.4778 0.477 0.4764
1.45 1 0.5 0.491 0.486 0.484 0.4822 0.4809 0.48 0.4792 0.4786
1.50 1 0.5 0.492 0.487 0.485 0.4839 0.4828 0.4819 0.4812 0.4807

 

 

100

Table 6.9. Gauss-Galerkin Finite Element Method: Changes of the exact total mo-
ment M i (t) as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5 M6 M7 M3 M9
0.00 l 0.5 0.333 0.25 0.2 0.166 0.1429 0.125 0.1111 0.1
0.05 1 0.5 0.349 0.273 0.228 0.198 0.1768 0.1607 0.1481 0.1381
0.10 1 0.5 0.363 0.295 0.254 0.227 0.2076 0.193 0.1816 0.1725
0.15 1 0.5 0.376 0.314 0.277 0.253 0.2354 0.2222 0.2119 0.2037
0.20 1 0.5 0.388 0.332 0.298 0.276 0.2606 0.2486 0.2393 0.2319
0.25 1 0.5 0.398 0.348 0.318 0.297 0.2834 0.2726 0.2641 0.2574
0.30 1 0.5 0.408 0.362 0.335 0.317 0.304 0.2942 0.2866 0.2805
0.35 1 0.5 0.417 0.375 0.351 0.334 0.3226 0.3138 0.3069 0.3014
0.40 1 0.5 0.425 0.387 0.365 0.350 0.3395 0.3315 0.3253 0.3203
0.45 1 0.5 0.432 0.398 0.378 0.364 0.3548 0.3475 0.3419 0.3374
0.50 1 0.5 0.438 0.408 0.389 0.377 0.3686 0.362 0.3569 0.3528
0.55 1 0.5 0.444 0.416 0.400 0.389 0.3811 0.3752 0.3706 0.3669
0.60 1 0.5 0.449 0.424 0.409 0.399 0.3924 0.3871 0.3829 0.3795
0.65 1 0.5 0.454 0.431 0.418 0.409 0.4027 0.3978 0.394 0.391
0.70 1 0.5 0.458 0.438 0.426 0.417 0.4119 0.4075 0.4041 0.4014
0.75 1 0.5 0.462 0.444 0.433 0.425 0.4203 0.4163 0.4132 0.4107
0.80 l 0.5 0.466 0.449 0.439 0.432 0.4279 0.4243 0.4215 0.4192
0.85 1 0.5 0.469 0.454 0.445 0.439 0.4348 0.4315 0.429 0.4269
0.90 1 0.5 0.472 0.458 0.450 0.444 0.441 0.438 0.4357 0.4339
0.95 1 0.5 0.475 0.462 0.455 0.450 0.4466 0.4439 0.4418 0.4402
1.00 1 0.5 0.477 0.466 0.459 0.454 0.4517 0.4492 0.4474 0.4459
1.05 1 0.5 0.479 0.469 0.463 0.459 0.4563 0.4541 0.4524 0.451
1.10 1 0.5 0.481 0.472 0.466 0.463 0.4604 0.4584 0.4569 0.4557
1.15 1 0.5 0.483 0.474 0.469 0.466 0.4642 0.4624 0.461 0.4599
1.20 1 0.5 0.484 0.477 0.472 0.469 0.4676 0.466 0.4647 0.4637
1.25 1 0.5 0.486 0.479 0.475 0.472 0.4707 0.4692 0.4681 0.4672
1.30 1 0.5 0.487 0.481 0.477 0.475 0.4735 0.4721 0.4711 0.4703
1.35 1 0.5 0.488 0.483 0.479 0.477 0.476 0.4748 0.4739 0.4731
1.40 1 0.5 0.489 0.484 0.481 0.479 0.4783 0.4772 0.4764 0.4757
1.45 1 0.5 0.490 0.486 0.483 0.481 0.4803 0.4794 0.4786 0.478
1.50 1 0.5 0.491 0.487 0.485 0.483 0.4822 0.4813 0.4806 0.4801

 

 

101

Table 6.10. Gauss-Galerkin Finite Element Method: Changes of the errors between
the exact total moment M 2'(t) and their approximation Mn(t) as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time EMO EM1 EM2 EM3 EM4 EM5
0.00 2.2e-16 5.6e-17 -5.6e-17 2.8e-17 5.6e-17 2.8e-17
0.05 1.1e-16 1.1e-16 -000015 -000023 -000027 -000031
0.10 4.4e-16 5e-16 -000028 -000041 -00005 -000055
0.15 4.4e-l6 1.7e-16 -000037 -0.00056 -0.00067 -0.00075
0.20 7.8e-16 5e-16 -000045 -000068 -000081 -00009
0.25 4.4e-16 5e-16 -000051 -000077 -000092 -0.001
0.30 7.8e-16 6.7e-16 -000055 -000083 -0001 -00011
0.35 5.6e-16 5.6e-16 -0.00059 -000088 -00011 -00012
0.40 8.9e-16 5.6e-16 -00006 -0.00091 -00011 -00012
0.45 1.1e-15 6.19-16 -000062 -000092 -00011 -00012
0.50 1e-15 7.2e-16 -000062 -000093 —00011 -00012
0.55 1.6e-15 8.3e—16 -0.00062 -000092 -00011 -00012
0.60 1.6e-15 6.1e-16 -000061 -000091 -00011 -00012
0.65 1.4e—15 7.2e-16 -0.00059 -0.00089 -0.0011 -0.0012
0.70 1.8e-15 1.2e-15 -000058 -000087 -0001 -00012
0.75 1.3e—15 le-15 -000056 —000084 -0001 -00011
0.80 1.7e—15 8.3e-16 ~000054 -000081 -000097 -00011
0.85 2.1e-15 1.2e—15 -000052 -0.00078 —0.00094 -0001
0.90 2.1e-15 8.9e-16 -00005 -000075 -00009 -0001
0.95 2.3e-15 1.1e-15 -000048 -000071 -000086 -000095
1.00 2.6e-15 1.4e-15 -0.00045 -000068 -000081 -000091
1.05 2.6e-15 1.4e-15 -000043 -000064 -000077 -000086
1.10 2.7e—15 1.3e-15 -000041 -000061 -000073 -000081
1.15 2.7e-15 1.2e-15 -000039 -000058 -000069 -000077
1.20 3.3e-15 1.4e-15 -000036 -000055 -000065 -000073
1.25 3e-15 1.4e-15 -0.00034 -000051 -000062 -000068
1.30 3.2e-15 1.4e-15 -000032 -000048 -000058 -000064
1.35 3.6e-15 1.7e-15 -00003 -000045 -000054 -00006
1.40 3.8e-15 1.7e-15 -000028 -000043 -000051 -000057
1.45 3.8e—15 1.6e-15 -000027 -00004 -0.00048 -0.00053
1.50 4.1e-15 1.9e-15 -000025 -000037 -000045 -00005

 

 

102

Table 6.11. Gauss-Galerkin Finite Element Method: Changes of the relative errors
between the exact total moment M ’(t) and their approximation Mn(t) as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time REMO REMI REM2 REM3 REM4
0.00 1.6867e-16 1.6867e-16 1.6867e-16 1.6867e-16 1.6867e—16
0.05 -000027869 -000027869 -000027869 -000027869 -000027869
0.10 -000054912 -000054912 -000054912 -000054912 -000054912
0.15 -000079 -0.00079 -0.00079 -000079 -000079
0.20 -000099243 -000099243 -000099243 -0.00099243 -0.00099243
0.25 -00011547 -00011547 -00011547 -0.0011547 -0.0011547
0.30 -00012788 -00012788 -00012788 -00012788 -00012788
035 -00013688 -00013688 -00013688 -00013688 -00013688
0.40 -00014291 -00014291 --00014291 -00014291 -00014291
0.45 -00014642 -00014642 -00014642 -00014642 -00014642
0.50 -00014783 -00014783 -00014783 -00014783 -00014783
0.55 -00014753 -00014753 -00014753 -00014753 -00014753
0.60 -0.0014584 -00014584 -00014584 -00014584 -00014584
0.65 -00014305 -00014305 -0.0014305 -0.0014305 -00014305
0.70 -00013939 -00013939 -00013939 -00013939 -00013939
0.75 -0.0013508 -0.0013508 -00013508 -00013508 -00013508
0.80 -00013028 -00013028 -00013028 -00013028 -00013028
0.85 -00012513 -00012513 -00012513 -00012513 -00012513
0.90 -00011975 -00011975 -00011975 -00011975 -00011975
0.95 -00011424 -00011424 -00011424 -00011424 -00011424
1.00 -0.0010867 -0.0010867 -00010867 -00010867 -00010867
1.05 -00010312 -00010312 -00010312 -00010312 -00010312
1.10 -000097627 -000097627 -000097627 -000097627 -000097627
1.15 -000092237 -000092237 -000092237 -000092237 -000092237
1.20 -000086983 -000086983 -000086983 -000086983 -000086983
1.25 -000081888 -0.00081888 -0.00081888 -000081888 -000081888
1.30 -00007 6971 -0.00076971 -000076971 -000076971 -000076971
1.35 -000072245 -000072245 -0.00072245 -0.00072245 -000072245
1.40 -000067719 -000067719 -000067719 -000067719 -000067719
1.45 -000063398 -000063398 -0.00063398 -0.00063398 -000063398
1.50 -000059284 -000059284 -000059284 -000059284 -000059284

 

 

 

 

 

 

 

 

103

and initial conditions
2 7T3! .
u(x,y,O) = 7rcos (—2—) sm(7r:r) on Q. (6.55)

Case I p = 2, q = 2

This implies that both :1: = 0 and :1: = 1 are natural boundaries. In this example,
for fixed y, one can show that the density u(x, y, 00) = 0, Va: 6 (0,1), is a steady
state solution and escaping to the boundary a: = 0 and :c = 1 as 1: becomes arbitrary
large and u will “pile” near boundary a: = O and a: = 1 with a Dirac-delta function
singularity formed at a: = 0' and a: = 1‘. Our numerical result will show this.

We apply the Gauss-Galerkin Finite Element method using finite element ap—
proximations in the y-direction and Gauss-Galerkin approximations in the x-direction.
We divided y E [0, 1] into four equal subintervals. Then we get five grid lines
y, = jh, j = 0,1,---,4,h = 0.25. In the x—direction we use three nodes labeled
by *, + and x for each grid line y = y,-. We only discuss the results along grid
y = 0.5. The results along y = 0.25 and y = 0.75 are similar. We use *, + and x to
indicate the nodes moving to the boundary x = O , interior point and the boundary
a: = 1. Figure 6.3 showes the movement of the three nodes as t increases. Table 6.12
shows the changes of nodes *, + and x as t increases. Table 6.13 shows the changes
of the weights at the three nodes *, + and x as t increases. Table 6.14 shows the
changes of the ith “moment” m§,(y, t) at y = 0.5 as t increases. Table 6.15 shows the

changes of the “total moment” Mf, (t) as t increases.

(1) Integrating (6.49) over $2, integrating by parts and using boundary condition
(6.53), (6.54) we have % fol f01u(.r, y, t)d$dy = fol fg((x2(1—a:)2u)u+uyy)dxdy =
0 This shows that the total probability is conserved. Therefore, the rate of
change of the exact 0th total moment £20,451 2 0 So, the exact 0th total

moment M°(t) = M°(0) = fol fol u(x,y,0)d:rdy = 1.

104

(2) Multiplying (6.49) by a: and integrating over $2, integrating by parts and
using boundary condition (6.53), (6.54) we have %f01f01u(x,y,t)xdxdy =
fol f01(($2(1 -— :r)2u)mx + uyyx)d:rdy = 0 This means the rate of change of
the exact lst total moment £491 = 0 So, the exact lst total moment

M1(t) = M1(0) = fol f0l u(x,y, O)a:dxdy = 0.5.

(3) Table 6.15 shows that M30) = 1 = M°(t) for t Z 1. This means the total
mass = 1. Table 6.13 shows that is equal to the sum of the two weights at the

boundaries :1: = 0 and :r = 1.

(4) Table 6.15 shows that M,1,(t) = 0.5 = M1(t) for t 2 1. This shows that the

approximation of the lst total moment equals the exact lst total moment.
(5) At t = 30, the solution reaches the steady state based on the tolerance chosen.

(6) The solutinon becomes uniform in y as t increases based on observations of the

numerical results along the other y-grid lines.

(7) The solution approaches zero in the interior and “piles” up near the boundaries

2r = 0 and :2: = 1 as t increases. The steady state solution is %(6(x) + 6(23 — 1)).
(8) The weight for the middle node tends to zero as t increases.
(9) The weights at the other two nodes tend to 0.5 as t increases.

(10) Table (6.15) shows that the ith “total moments” of the steady state solution are
all about 0.5 for i from 1 to 5. Since the weight tends to zero for the interior
node and 0.5 for the other two points which approach to x = 0 and a: = 1,
the contribution of u to the ith total moment is the contribution of these two
weights to the ith total moment. The total moment contributed by the weight
at a: = 0 is zero and the total moment contributed by the weight at :1: = 1 is 0.5

which is equal to the weight at :r = 1.

105

We now discuss how the movement of nodes depends on the parameters p and q. We
know that the degree of singularities of (6.52) depend on the parameters p and q. If
p < q(or p > q), the singularity at :1: = 0 will be less (or greater) than the singularity
at :1: = 1. The movement of nodes to boundary a: = 0 should be faster (or slower)
than the movement of nodes to boundary 2: = 1. Following Examples shall show that.
In the following Case II, Case III, and case IV, we use the same method as in Case I
to get the numerical results.

Case II p < q

We take p = 1 and q = 2 as an example. The numerical results are in Figure 6.4 and
Tables 6.16, 6.17, 6.18 and 6.19. We observe that the nodes move to the boundary
:1: = 0 faster than to the boundary a: = 1.

Case III p > q

We take p = 2 and q = 1 as an example. The numerical results are in Figure 6.5 and
Tables 6.20, 6.21, 6.22 and 6.3. We observe that the nodes move to the boundary
a: = 0 more slowly than to the boundary a: = 1.

Case IV Fixing p and changing q

Tables 6.24 6.25 6.26 and 6.27 show how the movement of nodes to boundary
x = 1 depends on the parameter q. Nodes with smaller parameter q move to the

boundary :1: = 1 faster than those with larger q.

106

Model II. Let‘ p = 1 and q = 1. This implies that both :1: 2 0 and :1: = 1 are exit

boundaries. We consider the following initial -boundary problem:
11, = (x(1 — x)u)“. + aw in (0,T) x Q (6.56)
with the boundary conditions
ii_r1(1)xu(x,y,t) = 1333(1 — x)u(x, y,t) = 0 on (0,T) x (0,1), (6.57)

uy(:c,0,t) = uy(:r,1,t) = 0 on (0,T) x (0,1), (6.58)

and initial conditions
u(x, y, 0) = 7r 0.55 — 0.1 cos2 (gfl :rsin(7ra:) on {2. (6.59)
By (6.50),
P(x, y, t) = —(:1:(1-— x)u); = —(1 — x)u + mu — x(1 — x)ux. (6.60)
Using the boundary condition (6.57) we have

P(O,y,t)=-U(0.y.t) and P(1.y.t)=U(1.y.t). (6-61)

107

Table 6.12. Gauss—Galerkin Finite Element Method: Changes of the nodes *, + and
x at y = 0.5 with p = 2 and q = 2 as t increases

 

time 1:1 2:2 1:3

0.00 01776790146 0.5 08223209854
0.20 01569951415 0.5 08430048585
0.40 01387971313 0.5 08612028687
0.60 0124213992 0.5 0875786008
0.80 08875133494 0.5 01124866506
1.00 01028788815 0.5 08971211185
2.00 007248655953 0.5 09275134405
3.00 005589220564 0.5 09441077944
4.00 004523506173 0.5 09547649383
5.00 003777675115 0.5 09622232489
6.00 003227123193 0.5 09677287681
7.00 002805416464 0.5 09719458354
8.00 002473328072 0.5 09752667193
9.00 09779398506 0.5 002206014938
10.0 001986926287 0.5 09801307371
11.0 001804605226 0.5 09819539477
12.0 001650877966 0.5 09834912203
13.0 001519767562 0.5 09848023244
14.0 001406812019 0.5 09859318798
15.0 00130862042 0.5 09869137958
16.0 001222575257 0.5 09877742474
17.0 001146627805 0.5 0988533722
18.0 001079154418 0.5 09892084558
19.0 001018853736 0.5 0.9898114626
20.0 0009646719368 0.5 09903532806
21.0 0009157475852 0.5 09908425241
22.0 0008713703905 0.5 09912862961
23.0 0008309499947 0.5 09916905001
24.0 0007939920837 0.5 09920600792
25.0 000760079916 0.5 09923992008
26.0 0007288599021 0.5 0992711401
27.0 0007000302461 0.5 09929996975
28.0 0006733319251 0.5 09932666807
29.0 0006485414691 0.5 09935145853
30.0 0006254651387 0.5 09937453486

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

108

Table 6.13. Gauss-Galerkin Finite Element Method: Changes of the weights at nodes
at, + and x at y = 0.5 with p = 2 and q = 2 as t increases

 

time

w1

w2

w3

 

0.00

02279202041

05441595918

0.2279202041

 

0.20

02708900461

04582199078

02708900461

 

0.40

02993447133

04013105734

0.2993447133

 

0.60

03216170199

03567659601

0.3216170199

 

0.80

03402289554

03195420892

0.3402289554

 

1.00

0356218263

0287563474

0.356218263

 

2.00

04112542944

01774914111

0.4112542944

 

3.00

04421675514

01156648973

0.4421675514

 

4.00

04606419939

007871601222

0.4606419939

 

5.00

04721989331

005560213382

0.4721989331

 

6.00

04797123027

004057539469

0.4797123027

 

7.00

0484765015

003046997007

0.484765015

 

8.00

04882673879

002346522416

0.4882673879

 

9.00

04907622074

001847558527

0.4907622074

 

10.0

04925835524

001483289517

0.4925835524

 

11.0

04939430195

00121139611

0.4939430195

 

12.0

04949781889

001004362224

0.4949781889

 

13.0

04957807059

0008438588265

0.4957807059

 

14.0

 

04964129983

0007174003459

0.4964129983

 

15.0

04969184809

0006163038117

0.4969184809

 

16.0

049732793

0005344140008

0.49732793

 

17.0

04976635516

0004672896755

0.4976635516

 

18.0

04979416325

0.004116735089

0.497 9416325

 

19.0

04981742991

0003651401712

0.4981742991

 

20.0

04983707076

0003258584867

0.4983707076

 

21.0

04985378601

0002924279736

0.4985378601

 

22.0

0.498681 1764

0002637647259

0.4986811764

 

23.0

04988048972

0002390205651

0.4988048972

 

24.0

04989123752

000217524967

0.4989123752

 

25.0

0499006286

0001987428079

0.499006286

 

26.0

04990887838

0001822432463

0.4990887838

 

27.0

04991616173

0001676765466

0.4991616173

 

28.0

 

04992262168

0001547566318

0.4992262168

 

29.0

04992837609

0001432478197

0.4992837609

 

 

30.0

 

04993352268

 

0001329546435

 

0.4993352268

 

 

Table 6.14. Gauss-Galerkin Finite Element Method: Changes of the “moment”

109

mf,(y,t) at y = 0.5 with p = 2 and q = 2 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time m0 m1 m2 m3 m4 m5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
0.20 1 0.5 0.313742 0.220613 0.165612 0.129676
0.40 1 0.5 0.32811 0.242164 0.189855 0.154364
0.60 l 0.5 0.340834 0.261252 0.211579 0.176861
0.80 1 0.5 0.352182 0.278273 0.231117 0.197338
1.00 1 0.5 0.362355 0.293532 0.248751 0.215991
2.00 1 0.5 0.400328 0.350492 0.315467 0.287848
3.00 1 0.5 0.424419 0.386628 0.358529 0.335276
4.00 l 0.5 0.440532 0.410798 0.387702 0.367925
5.00 1 0.5 0.451771 0.427656 0.408265 0.391235
6.00 1 0.5 0.459894 0.43984 0.423259 0.408413
7.00 1 0.5 0.465946 0.448919 0.434518 0.421429
8.00 1 0.5 0.470578 0.455867 0.443191 0.431532
9.00 1 0.5 0.474206 0.461309 0.450024 0.439544
10.0 1 0.5 0.477106 0.465659 0.455513 0.446017
11.0 1 0.5 0.479466 0.469199 0.459999 0.451333
12.0 1 0.5 0.481416 0.472124 0.463721 0.455762
13.0 1 0.5 0.48305 0.474575 0.466849 0.459499
14.0 1 0.5 0.484436 0.476654 0.469511 0.462688
15.0 1 0.5 0.485624 0.478436 0.471799 0.465437
16.0 1 0.5 0.486652 0.479978 0.473784 0.467828
17.0 1 0.5 0.48755 0.481325 0.47552 0.469925
18.0 1 0.5 0.48834 0.48251 0.47705 0.471776
19.0 1 0.5 0.489039 0.483559 0.478408 0.473422
20.0 1 0.5 0.489663 0.484494 0.47962 0.474894
21.0 1 0.5 0.490222 0.485333 0.480709 0.476217
22.0 1 0.5 0.490726 0.486088 0.48169 0.477412
23.0 1 0.5 0.491182 0.486773 0.482581 0.478497
24.0 1 0.5 0.491596 0.487395 0.483391 0.479486
25.0 1 0.5 0.491975 0.487963 0.484131 0.48039
26.0 1 0.5 0.492322 0.488483 0.48481 0.481221
27.0 1 0.5 0.492641 0.488962 0.485435 0.481986
28.0 1 0.5 0.492935 0.489403 0.486012 0.482692
29.0 1 0.5 0.493208 0.489812 0.486546 0.483347
30.0 1 0.5 0.49346 0.490191 0.487042 0.483955

 

 

 

Table 6.15. Gauss-Galerkin Finite Element Method: Changes of the “total moment”

110

Mf,(t) with p = 2 and q = 2 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
0.20 1 0.5 0.313742 0.220613 0.165612 0.129676
0.40 1 0.5 0.32811 0.242164 0.189855 0.154364
0.60 1 0.5 0.340834 0.261252 0.211579 0.176861
0.80 1 0.5 0.352182 0.278273 0.231117 0.197338
1.00 1 0.5 0.362355 0.293532 0.248751 0.215991
2.00 1 0.5 0.400328 0.350492 0.315467 0.287848
3.00 1 0.5 0.424419 0.386628 0.358529 0.335276
4.00 1 0.5 0.440532 0.410798 0.387702 0.367925
5.00 1 0.5 0.451771 0.427656 0.408265 0.391235
6.00 1 0.5 0.459894 0.43984 0.423259 0.408413
7.00 1 0.5 0.465946 0.448919 0.434518 0.421429
8.00 1 0.5 0.470578 0.455867 0.443191 0.431532
9.00 1 0.5 0.474206 0.461309 0.450024 0.439544
10.0 1 0.5 0.477106 0.465659 0.455513 0.446017
11.0 1 0.5 0.479466 0.469199 0.459999 0.451333
12.0 1 0.5 0.481416 0.472124 0.463721 0.455762
13.0 1 0.5 0.48305 0.474575 0.466849 0.459499
14.0 1 0.5 0.484436 0.476654 0.469511 0.462688
15.0 1 0.5 0.485624 0.478436 0.471799 0.465437
16.0 1 0.5 0.486652 0.479978 0.473784 0.467828
17.0 1 0.5 0.48755 0.481325 0.47552 0.469925
18.0 1 0.5 0.48834 0.48251 0.47705 0.471776
19.0 1 0.5 0.489039 0.483559 0.478408 0.473422
20.0 1 0.5 0.489663 0.484494 0.47962 0.474894
21.0 1 0.5 0.490222 0.485333 0.480709 0.476217
22.0 1 0.5 0.490726 0.486088 0.48169 0.477412
23.0 1 0.5 0.491182 0.486773 0.482581 0.478497
24.0 1 0.5 0.491596 0.487395 0.483391 0.479486
25.0 1 0.5 0.491975 0.487963 0.484131 0.48039
26.0 1 0.5 0.492322 0.488483 0.48481 0.481221
27 .0 1 0.5 0.492641 0.488962 0.485435 0.481986
28.0 1 0.5 0.492935 0.489403 0.486012 0.482692
29.0 1 0.5 0.493208 0.489812 0.486546 0.483347
30.0 1 0.5 0.49346 0.490191 0.487042 0.483955

 

 

 

111

Table 6.16. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and
x at y = 0.5 with p =1 and q = 2 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time :01 2:2 :53

0.00 0.1776790146 0.5 0.8223209854
0.20 004737482572 04409938921 08241783121
0.40 002403084443 04448604727 08476643036
0.60 001482958065 04550658882 08673940809
0.80 001008567793 04651212708 08830828805
1.00 0.007289118074 04740494633 08956991781
2.00 0002250069707 05051335296 09335385097
3.00 00009863745097 0524526958 09522693971
4.00 00005184071064 05382862795 09632779238
5.00 00003076309523 05484473913 09704182539
6.00 00001992931646 05560949596 09753690413
7.00 00001378318697 05619588701 09789751276
8.00 00001001977718 05665471543 09817040036
9.00 7.572609049e-05 05702100595 09838329178
10.0 5.902989083e-05 0573188811 09855354713
11.0 4.718510394e-05 05756514259 09869252876
12.0 3.850700958e—05 05777170269 09880795345
13.0 3.197534068e-05 05794717514 09890522877
14.0 2.694566813e-05 05809791037 09898824725
15.0 2.299607804e-05 05822867591 09905987671
16.0 1.984166688e—05 0.5834311111 09912227338
17.0 1.728481074e—05 05844403654 09917708763
18.0 1.518516201e-05 05853366859 09922560286
19.0 1 .344095633e-05 05861377084 09926883123
20.0 1.197703012e-05 05868576267 0993075812

21.0 1.073694277e-O5 05875079836 09934250589
22.0 9.677678823e—06 05880982562 09937413846
23.0 8.7660117286—06 05886362931 09940291821
24.0 7 .975961631e—06 05891286474 09942921025
25.0 7.286987989e-06 05895808303 09945332042

 

 

 

 

 

 

Table 6.17. Gauss-Galerkin Finite Element Method: Changes of the weights at nodes

112

*, + and x at y=05 withp= 1 and q: 2 as t increases

 

time

wl

W2

013

 

0.00

0.2279202041

0.5441595918

0.2279202041

 

0.20

0208671766

04229852476

03683429864

 

0.40

02621331027

0327113844

04107530533

 

0.60

03051459394

02600779851

04347760755

 

0.80

03380120395

02105440153

04514439451

 

1.00

03633954976

0172785513

04638189894

 

2.00

04313189796

007435985001

04943211704

 

3.00

 

04583094813

003807751325

05036130054

 

4.00

04714473737

002208861808

05064640082

 

5.00

04787955491

00140616475

05071428034

 

6.00

04833461619

0009598700584

05070551375

 

7.00

04863870524

000690942014

05067035274

 

8.00

0488539708

0005182539881

05062777521

 

9.00

0490132959

0004015963892

05058510771

 

10.0

04913542797

0003194894872

05054508254

 

11.0

04923172628

0002597202503

05050855347

 

12.0

0493094267

0002149729241

05047560038

 

13.0

04937333422

0001806685886

05044599719

 

14.0

04942675308

0001538322743

05041941464

 

15.0

04947202406

000132468135

0503955078

 

16.0

04951084827

0001151994663

05037395226

 

17.0

04954448965

0001010532885

05035445707

 

18.0

04957390564

00008932708089

05033676728

 

19.0

04959983405

00007950388424

05032066207

 

20.0

04962285206

00007119678796

05030595115

 

21.0

04964341739

0.000641 1177727

05029247083

 

22.0

04966189742

00005802234192

05028008024

 

23.0

04967859031

00005275179256

0502686579

 

24.0

04969374041

00004816073189

05025809886

 

 

25.0

 

 

04970754986

 

00004413803845

 

0502483121

 

 

 

Table 6.18. Gauss-Galerkin Finite Element Method: Changes of the “moment”

113

mj,(y, t) at y = 0.5 with p = 1 and q = 2 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time mo m1 m2 m3 m4 m5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
0.20 l 0.5 0.332933 0.242511 0.185955 0.147129
0.40 1 0.5 0.360028 0.278982 0.22488 0.185462
0.60 1 0.5 0.381039 0.308246 0.257264 0.218551
0.80 1 0.5 0.397635 0.332077 0.284396 0.247027
1.00 l 0.5 0.410959 0.351707 0.307262 0.271535
2.00 1 0.5 0.449774 0.411751 0.380279 0.352931
3.00 1 0.5 0.467161 0.440382 0.417012 0.395875
4.00 1 0.5 0.476351 0.456138 0.437923 0.421054
5.00 1 0.5 0.481812 0.465774 0.451017 0.437138
6.00 1 0.5 0.485353 0.472153 0.459832 0.448121
7.00 1 0.5 0.487803 0.476637 0.466104 0.456017
8.00 1 0.5 0.489585 0.479937 0.470765 0.46193
9.00 1 0.5 0.490933 0.482456 0.474348 0.466503
10.0 1 0.5 0.491984 0.484435 0.47718 0.470135
11.0 1 0.5 0.492825 0.486027 0.479469 0.473083
12.0 1 0.5 0.493511 0.487334 0.481355 0.475518
13.0 1 0.5 0.494082 0.488424 0.482933 0.477563
14.0 1 0.5 0.494563 0.489347 0.484272 0.479301
15.0 1 0.5 0.494973 0.490136 0.485422 0.480796
16.0 1 0.5 0.495328 0.49082 0.486419 0.482095
17.0 1 0.5 0.495636 0.491417 0.487291 0.483233
18.0 1 0.5 0.495908 0.491943 0.48806 0.484238
19.0 1 0.5 0.496148 0.492409 0.488744 0.485132
20.0 1 0.5 0.496362 0.492826 0.489355 0.485932
21.0 1 0.5 0.496554 0.4932 0.489904 0.486652
22.0 1 0.5 0.496727 0.493537 0.490401 0.487303
23.0 1 0.5 0.496884 0.493843 0.490851 0.487895
24.0 1 0.5 0.497027 0.494122 0.491262 0.488435
25.0 1 0.5 0.497158 0.494378 0.491638 0.488929

 

 

_F“

114

Table 6.19. Gauss-Galerkin Finite Element Method: Changes of the “total moment”
Mfi,(t) at y = 0.5 with p = 1 and q = 2 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5
0.00 l 0.5 0.297358 0.196036 0.138456 0.102747
0.20 1 0.5 0.332933 0.242511 0.185955 0.147129
0.40 1 0.5 0.360028 0.278982 0.22488 0.185462
0.60 1 0.5 0.381039 0.308246 0.257264 0.218551
0.80 1 0.5 0.397635 0.332077 0.284396 0.247027
1.00 1 0.5 0.410959 0.351707 0.307262 0.271535
2.00 l 0.5 0.449774 0.411751 0.380279 0.352931
3.00 l 0.5 0.467161 0.440382 0.417012 0.395875
4.00 1 0.5 0.476351 0.456138 0.437923 0.421054
5.00 1 0.5 0.481812 0.465774 0.451017 0.437138
6.00 1 0.5 0.485353 0.472153 0.459832 0.448121
7.00 l 0.5 0.487803 0.476637 0.466104 0.456017
8.00 1 0.5 0.489585 0.479937 0.470765 0.46193
9.00 1 0.5 0.490933 0.482456 0.474348 0.466503
10.0 1 0.5 0.491984 0.484435 0.47718 0.470135
11.0 1 0.5 0.492825 0.486027 0.479469 0.473083
12.0 1 0.5 0.493511 0.487334 0.481355 0.475518
13.0 1 0.5 0.494082 0.488424 0.482933 0.477563
14.0 1 0.5 0.494563 0.489347 0.484272 0.479301
15.0 1 0.5 0.494973 0.490136 0.485422 0.480796
16.0 1 0.5 0.495328 0.49082 0.486419 0.482095
17.0 1 0.5 0.495636 0.491417 0.487291 0.483233
18.0 1 0.5 0.495908 0.491943 0.48806 0.484238
19.0 1 0.5 0.496148 0.492409 0.488744 0.485132
20.0 1 0.5 0.496362 0.492826 0.489355 0.485932
21.0 1 0.5 0.496554 0.4932 0.489904 0.486652
22.0 1 0.5 0.496727 0.493537 0.490401 0.487303
23.0 1 0.5 0.496884 0.493843 0.490851 0.487895
24.0 1 0.5 0.497027 0.494122 0.491262 0.488435
25.0 1 0.5 0.497158 0.494378 0.491638 0.488929

 

 

 

 

 

 

 

 

 

Table 6.20. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

115

x at y = 0.5 withp= 2 and q =1 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 11:1 51:2 :133

0.00 0.1776790146 0.5 0.8223209854
0.20 01758216879 05590061079 09526251743
0.40 01523356964 05551395273 09759691556
0.60 01326059191 05449341118 09851704193
0.80 01169171195 05348787292 09899143221
1.00 01043008219 05259505367 09927108819
2.00 006646149033 04948664704 09977499303
3.00 004773060294 0475473042 09990136255
4.00 003672207624 04617137205 09994815929
5.00 00295817461 04515526087 0999692369
6.00 002463095868 04439050404 09998007068
7.00 002102487236 0.4380411299 09998621681
8.00 001829599644 04334528457 09998998022
9.00 001616708221 04297899405 09999242739
10.0 001446452871 0426811189 09999409701
11.0 001307471238 04243485741 09999528149
12.0 001192046553 04222829731 0999961493
13.0 001094771232 04205282486 09999680247
14.0 0.01011752754 04190208963 09999730543
15.0 0009401232879 04177132409 09999770039
16.0 0008777266222 04165688889 09999801583
17.0 0008229123669 04155596346 09999827152
18.0 0007743971397 04146633141 09999848148
19.0 000731168767 04138622916 0999986559
20.0 0006924188018 04131423733 0999988023
21.0 0006574941083 04124920164 09999892631
22.0 0006258615421 0.4119017438 09999903223
23.0 0005970817947 04113637069 0999991234
24.0 0005707897548 04108713526 0999992024
25.0 0005466795779 04104191697 0999992713
26.0 0005244932026 04100024907 09999933173
27.0 0005040114249 04096173381 09999938502
28.0 0004850468907 04092603019 09999943223
29.0 0004674385462 04089284431 09999947426
30.0 0004510472031 04086192151 09999951184

 

 

 

 

Table 6.21. Gauss-Galerkin Finite Element Method: Changes of the weights at nodes

116

*, + and x at y =05 withp= 2 and q=1 as t increases

 

time

w1

012

013

 

0.00

0.2279202041

0.5441595918

02279202041

 

0.20

0.3683429864

0.4229852476

0.208671766

 

0.40

04107530533

0.327113844

0.2621331027

 

0.60

04347760755

0.2600779851

0.3051459394

 

0.80

0.4514439451

0.2105440153

0.3380120395

 

1.00

0.4638189894

0.172785513

0.3633954976

 

2.00

0.4943211704

0.07435985001

0.4313189796

 

3.00

0.5036130054

0.03807751325

04583094813

 

4.00

0.5064640082

0.02208861808

0.4714473737

 

5.00

0.5071428034

0.0140616475

04787955491

 

6.00

0.5070551375

0.009598700584

04833461619

 

7.00

0.5067035274

0.00690942014

0.4863870524

 

8.00

0.5062777521

0.005182539881

0.488539708

 

9.00

0.5058510771

0.004015963892

0.490132959

 

10.0

05054508254

0.003194894872

0.4913542797

 

11.0

0.5050855347

0.002597202503

0.4923172628

 

12.0

0.5047560038

0.002149729241

0.493094267

 

13.0

0.5044599719

0.001806685886

0.4937333422

 

14.0

0.5041941464

0.001538322743

0.4942675308

 

15.0

0.503955078

0.00132468135

0.4947202406

 

16.0

0.5037395226

0.001151994663

0.4951084827

 

17.0

0.5035445707

0.001010532885

0.4954448965

 

18.0

0.5033676728

00008932708089

0.4957390564

 

19.0

0.5032066207

00007950388424

0.4959983405

 

20.0

0.5030595115

0.0007119678796

0.4962285206

 

21.0

0.5029247083

0.0006411177727

0.4964341739

 

22.0

0.5028008024

00005802234192

0.4966189742

 

23.0

0.502686579

00005275179256

0.4967859031

 

24.0

0.5025809886

00004816073189

0.4969374041

 

25.0

0.502483121

00004413803845

0.4970754986

 

26.0

05023921849

00004059428479

04972018722

 

27.0

05023074899

00003745686933

04973179415

 

28.0

05022284312

00003466637192

04974249051

 

29.0

05021544778

0000321737949

04975237843

 

 

30.0

 

 

05020851616

 

00002993845278

 

04976154538

 

 

117

Table 6.22. Gauss—Galerkin Finite Element Method: Changes of the “moment”
m§,(y, t) at y = 0.5 with p = 2 and q = 1 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 7710 m1 m2 m3 m4 777.5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
0.20 1 0.5 0.332933 0.256287 0.213507 0.186861
0.40 l 0.5 0.360028 0.301101 0.269119 0.249395
0.60 1 0.5 0.381039 0.33487 0.310512 0.295697
0.80 1 0.5 0.397635 0.360828 0.341898 0.330535
1.00 1 0.5 0.410959 0.381172 0.366192 0.357303
2.00 1 0.5 0.449774 0.437571 0.431919 0.428696
3.00 1 0.5 0.467161 0.461102 0.458453 0.456979
4.00 l 0.5 0.476351 0.472914 0.471475 0.47069
5.00 1 0.5 0.481812 0.479662 0.478792 0.478324
6.00 1 0.5 0.485353 0.483904 0.483334 0.48303
7.00 1 0.5 0.487803 0.486771 0.486373 0.486163
8.00 l 0.5 0.489585 0.488818 0.488527 0.488374
9.00 1 0.5 0.490933 0.490343 0.490122 0.490006
10.0 1 0.5 0.491984 0.491517 0.491344 0.491255
11.0 1 0.5 0.492825 0.492447 0.492309 0.492237
12.0 1 0.5 0.493511 0.4932 0.493087 0.493028
13.0 1 0.5 0.494082 0.493821 0.493727 0.493678
14.0 1 0.5 0.494563 0.494341 0.494262 0.494221
15.0 1 0.5 0.494973 0.494783 0.494715 0.49468
16.0 1 0.5 0.495328 0.495163 0.495104 0.495074
17.0 1 0.5 0.495636 0.495492 0.495441 0.495415
18.0 1 0.5 0.495908 0.49578 0.495735 0.495712
19.0 1 0.5 0.496148 0.496035 0.495995 0.495975
20.0 1 0.5 0.496362 0.496261 0.496225 0.496207
21.0 1 0.5 0.496554 0.496463 0.496431 0.496415
22.0 1 0.5 0.496727 0.496645 0.496616 0.496602
23.0 1 0.5 0.496884 0.49681 0.496784 0.49677
24.0 1 0.5 0.497027 0.496959 0.496935 0.496923
25.0 1 0.5 0.497158 0.497095 0.497074 0.497063
26.0 1 0.5 0.497277 0.49722 0.4972 0.49719
27.0 1 0.5 0.497387 0.497335 0.497316 0.497307
28.0 1 0.5 0.497489 0.49744 0.497423 0.497415
29.0 1 0.5 0.497583 0.497538 0.497522 0.497514
30.0 1 0.5 0.497671 0.497629 0.497614 0.497607

 

 

 

 

 

 

 

 

118

Table 6.23. Gauss-Galerkin Finite Element Method: Changes of the “total moment”
Mfi,(t) at y = 0.5 with p = 2 and q = 1 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
0.20 1 0.5 0.332933 0.256287 0.213507 0.186861
0.40 1 0.5 0.360028 0.301101 0.269119 0.249395
0.60 1 0.5 0.381039 0.33487 0.310512 0.295697
0.80 1 0.5 0397635 0.360828 0.341898 0.330535
1.00 1 0.5 0.410959 0.381172 0.366192 0.357303
2.00 1 0.5 0.449774 0.437571 0.431919 0.428696
3.00 1 0.5 0.467161 0.461102 0.458453 0.456979
4.00 1 0.5 0.476351 0.472914 0.471475 0.47069
5.00 1 0.5 0.481812 0.479662 0.478792 0.478324
6.00 1 0.5 0.485353 0.483904 0.483334 0.48303
7.00 1 0.5 0.487803 0.486771 0.486373 0.486163
8.00 1 0.5 0.489585 0.488818 0.488527 0.488374
9.00 1 0.5 0.490933 0.490343 0.490122 0.490006
10.0 1 0.5 0.491984 0.491517 0.491344 0.491255
11.0 1 0.5 0.492825 0.492447 0.492309 0.492237
12.0 1 0.5 0.493511 0.4932 0.493087 0.493028
13.0 1 0.5 0.494082 0.493821 0.493727 0.493678
14.0 1 0.5 0.494563 0.494341 0.494262 0.494221
15.0 1 0.5 0.494973 0.494783 0.494715 0.49468
16.0 1 0.5 0.495328 0.495163 0.495104 0.495074
17.0 1 0.5 0.495636 0.495492 0.495441 0.495415
18.0 1 0.5 0.495908 0.49578 0.495735 0.495712
19.0 1 0.5 0.496148 0.496035 0.495995 0.495975
20.0 1 0.5 0.496362 0.496261 0.496225 0.496207
21.0 1 0.5 0.496554 0.496463 0.496431 0.496415
22.0 1 0.5 0.496727 0.496645 0.496616 0.496602
23.0 1 0.5 0.496884 0.49681 0.496784 0.49677
24.0 1 0.5 0.497027 0.496959 0.496935 0.496923
25.0 1 0.5 0.497158 0.497095 0.497074 0.497063
26.0 1 0.5 0.497277 0.49722 0.4972 0.49719
27.0 1 0.5 0.497387 0.497335 0.497316 0.497307
28.0 1 0.5 0.497489 0.49744 0.497423 0.497415
29.0 1 0.5 0.497583 0.497538 0.497522 0.497514
30.0 1 0.5 0.497671 0.497629 0.497614 0.497607

 

 

 

119

Integrating (6.56) over Q, integrating by parts and using boundary condition (6.57 ),
(6.58) we have

d_dt/ol fou(:r,ty,)d:1:dy 2/01/0((:1:(1 —:1:)u u)m+uyy)d:1:dy
l

=/0:( :1:((1—:1:)11.):|,,:Oaly+/0 uylizodx

=/:— P(x ,y,t) zody (6.62)
=/ -{P(1.y.t)-P(1 y. t)}dy

0 l
= -/ U(0.y.t)dy-/ U(l.y.t)dy-

0 0

Integrating (6.62) over 7' E (0, t), we have

1 1 1 1
/ / u(x,y,tmdy— / / u(x,y.0)dxdy
o o o o (6.63)

t 1 t 1
= -/ / u(0,y,7')dyd'r —/ / u(1,y,T)dydT
o o o 0

so,

[O/u (,xy,t)d:1:dy
2/09/0“ “(“110 W61?! ([14 (.7011. )dydT—fofu (1,y, )dyd’r (6.64)

=05—f / u(0,y,7‘)dydT—/ / u(1,y,'r)dydt:r.
o o o o

Multiplying by :1: and integrating (6.56) over 9, integrating by parts and using bound-
ary condition (6.57), (6.58) we have

ddt/o [0111 (:1: ,y,t):1:d:1:dy 2/0 [0(( :1:()ul—:1: )ma:+uyy:r)dxdy
1
:fo:( 2:( (1 —:1: u))x:1:|$:0dy+/ uy|;:0:rd:1:
o

= / —P<a: 11.6221..- .06
2/01—P(,1y,t)dy= -/0U( (1 ,yt,) d.y

(6.65)

120

Integrating (6.65) over 7' 6 (0, t), we have

1 1 1 1 t 1
ffu(x,y,t):1:d:1:dy—//u(x,y,O):1:d:cdy=—//u(1,y,'r)dyd'r. (6.66)
o o 0 o o 0

Thus,

1 1 1 1 t 1
f / u(m,y,t):1:d:1:dy =/ / u(:1:,y,0):1:d2:dy—/ / u(1,y,7')dyd7'
0 0 0 o o o

t 1 (6.67)
=0.2974— [0 / u(1,y,T)dydT.
0

where f6 f01u(0,y,1')dyd'r and fat fol u(l, y, T)dydT represent the fluxes across bound-
aries :1: = 0 and :1: = 1. In this example, a: = 0 and :c = 1 are exit boundaries. For
fixed y, one can show that the density u(x, y, 00) = 0, Vs 6 (0,1), is a steady state
solution and u will “pile” at :1: = 0 and :1: = 1 with a Dirac-delta function singularity
formed at z = 0 and z = 1. So, for steady state solutin, fol folu(:1:,y,t)d:1:dy = 0 in
the sense that the integration is only over the Open region 0. Therefore, by (6.64)
and (6.67), we have

t 1 t 1
f0 [0 u(0,y,T)dydT+/ / u(1,y,T)dydT= 0.5 (6.68)
o o

t 1
f / u(1,y,T)dydT=02974. (6.69)
0 0

Substituting (6.69) into (6.68), we obtain

t 1
f / u(0,y,T)dydT=0.2026. (6.70)
0 0

(6.69) and (6.70) show that the amounts of the fluxes leaking out of the boundaries
:1: = 0 and a: = 1 are 0.2026 and 0.2974 respectively. Our numerical result shall sup-
port this.

As in Model I, we apply the Gauss—Galerkin finite element method using finite el-

121

ement approximations in the y—direction and Gauss-Galerkin approximations in the
x-direction. We divide y E [0, 1] into four equal subintervals. Then we have five grid
lines y,- = jh, j = 0,1,--- , 4, h = 0.25. In the x-direction we use three nodes labeled
by 21:, + and x for each grid line y = y,-. We only discuss the results along grid y = 0.5.
The results along y = 0.25 and y = 0.75 are similar. Table 6.28 shows the changes
of nodes *, + and x as t increases. Table 6.29 shows the changes of the weights at
the three nodes ..., + and x as t increases. Table 6.30 shows the changes of the ith
“moment” mf,(y, t) at y = 0.5 as t increases. Table 6.31 shows changes of the “total

moment” M,‘,(t) as t increases.

(1) Table 6.31 shows that M2(t) = 0.5 and M; (t) = 0.297358 for t Z 0 This shows
that the approximation of the 0th and lst total moments equal the exact 0th

and lst total moments.
(2) At t =2 5, the solution reaches the steady state based on the tolerance chosen.
(3) The solutinon becomes uniform in y as t increases.

(4) The solution approaches zero in the interior and “piles” up at the boundary

:1: = 0 and a: = 1 as t increases.
(5) The weights for the middle nodes tend to zero as t increases.

(6) The weights at the other two nodes tend to 0.202642 and 0.297357 respectively
as t increases. They are the amounts of the fluxes leaking out from :1: = 0 and

33:1.

(7) The steady state solution shows that the ith “total moment” are all about
0.297358 for i from 1 to 5. Since the weight tends to zero for the interior

node , 0.202642 for the node approaching to :1: = 0 and 0.297358 for the node

approaching :1: = 1, the contribution of u to the total moments is that of those

 

122

two weights to the total moment. The total moment contributed by the weight
at :1: = 0 is zero and the total moment contributed by the weight at a: = 1 is

0.297358 which is equal to the weight at a: = 1.

123

6.4 Dependence of Solution upon Lower Order

Terms

We consider the following initial-boundary value problems:
1 2 .
u) = —(au)x + 5(1) 11)” + uyy 1n (0,T) x Q

with the boundary conditions

lim {an — $60920} = lim {an _1M}= 0 on (0,T) x (0,1)

 

3x :c—11 2 0:1:
nil/($107 t) : Uy($,1,t) : 0 on (0,T) X (011)

and initial conditions
u(x, y, 0) = 7rcos2 (lg) sin(7r:1:) on {2

where
a(x) = sa:(1— :1:)(h + (1 — 2h):1:) — 112: + u(l — :13)

b2(a:) = $330 — 11:)

CaseI s=0,,u=I/=0375

(6.71)

(6.72)

(6.73)

(6.74)

(6.75)

(6.76)

For this model, both :1: = 0 and :1: = 1 are entrance boundaries. The numerical

solution is expressed in Figure 6.6, Tables 6.32, 6.33, 6.34, and 6.35. We observe the

following facts:

(1) At t = 5, the solution reaches the steady state based on the tolerance chosen.

(2) Table 6.35 shows that M2(t) = 1,t 2 0;M,1,(t) = 05,t 2 0; Mflt) = 03125,t Z

5; M3(t) = 0.21875,t 2 5; M3(t) = 0.164062,t 2 5, M20) = 0.128906,t 2 5.

Table 6.24. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

124

x at y = 0.5 with p = 1 and q = 1.5 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 231 $2 (133

0.00 0.1776790146 0.5 0.8223209854
1.00 0007293214038 04974373111 09525687726
2.00 0001729665784 05197840348 0979205428
3.00 00005604709585 05350376191 09890444785
4.00 00002200237825 05471084894 09935435888
5.00 00001006076538 0556561809 09958664928
6.00 5.20099078e-05 05637753157 09971781213
7.00 2.9615875016-05 05692445438 09979737196
8.00 1.818914568-05 05734347156 09984853627
9.00 1.185755384e-05 05767034381 0998830589
10.0 8.107449522e-06 05793039239 09990729698
11.0 5.7623219276-06 0.5814116194 0999248879
12.0 4.2285666186-06 05831487323 09993801606
13.0 3.18718017e-06 0584601736 09994804905
14.0 2.4573452926—06 05858329852 09995587449
15.0 1.931810867e-06 05868883096 09996208684
16.0 1 .5444299189-06 05878020086 09996709522
17.0 1253002659606 05886001862 0.999711881
18.0 1.029798929e-06 05893030166 09997457327
19.0 8.561 160685e-07 05899263102 09997740324
20.0 7.190453803e-07 05904826164 0999797919
21.0 6.094917483e-07 05909820126 09998182558
22.0 5.2092742386—07 05914326765 09998357065
23.0 4.485885282e-07 05918413091 09998507877
24.0 3.8894549618-07 05922134506 09998639063
25.0 3.39347036-07 0592553718 0999875386
26.0 2.977763748e-O7 05928659899 09998854866
27.0 2.6268163466-07 05931535445 09998944189
28.0 2.328558889e-07 0593419172 09999023551
29.0 2.073512635e-07 05936652628 09999094371
30.0 1.854165287e-07 05938938743 09999157823

 

 

Table 6.25. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

125

x at y = 0.5 with p = 1 and q = 2.5 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 21:1 1132 2:3

0.00 0.1776790146 0.5 0.8223209854
1.00 0006934697153 04474280303 08442717204
2.00 0002390166716 04815722684 08828764173
3.00 0.001194973711 05020708575 09051789036
4.00 00007074098499 05162853797 0919730071
5.00 00004640957196 05268758356 0930018803
6.00 00003268038883 05350669994 09377028371
7.00 00002423600226 05415689932 09436744707
8.00 00001869473545 05468414933 09484583795
9.00 00001487047648 0.551 1965095 09523838006
10.0 00001212280518 05548516757 09556679567
1 1.0 00001008288947 05579622653 09584600385
12.0 8.526779069e-05 05606414518 09608659536
13.0 7.312371114e—05 0562973402 09629629983
14.0 6.346080839e—05 0565021882 0964808936
15.0 5.564286602e-05 05668360216 09664478215
16.0 4.922533953e-05 05684542522 09679138568
17.0 4.389009955e-O5 05699070505 0969234012
18.0 3.940465289e—05 05712188853 09704298507
19.0 3.559590347e—05 05724096224 09715188278
20.0 3.233282601e-05 05734955559 09725152292
21.0 2.951475796e05 05744901743 09734308648
22.0 2.706332218e-O5 05754047389 09742755861
23.0 2.491674961e-05 05762487247 0975057679
24.0 2.302582074e-OS 05770301612 09757841653
25.0 2.135091924e—05 05777558979 09764610369
26.0 1.98598627e—05 05784318137 09770934395
27 .0 1 .85262848e—05 0579062983 0977685818
28.0 1.7 32841432e-05 05796538091 09782420332
29.0 1 .624814341e-05 05802081316 09787654554
30.0 1.527030942e-05 05807293142 09792590406

 

 

Table 6.26. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

126

xaty=05withp=landq=3astincreases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 21:1 :132 233

0.00 0.1776790146 0.5 0.8223209854
1.00 0006622257948 04211193674 08038513326
2.00 000233840896 0457402078 0837010961
3.00 0001249800175 04785285189 08591502431
4.00 00007922273888 04924933102 08746602438
5.00 00005508072973 05028776844 0.8862115773
6.00 00004068231039 0511067948 08952136158
7.00 00003138615836 05177367631 09024675006
8.00 00002502851745 05232846596 09084639723
9.00 00002048309658 05279787433 09135220107
10.0 00001711613512 05320071041 09178588005
11.0 00001454895413 0.5355063113 09216278052
12.0 0000125440224 05385778075 09249408108
13.0 00001094620382 05412984505 09278813437
14.0 9.650658451e-05 05437275232 09305131806
15.0 8.584437752e-05 0.5459114949 0932885936
16.0 7.695486689e—05 05478873207 09350388391
17.0 6.945832895e-05‘ 054968477 09370033543
18.0 6.307235147e-05 05513281031 09388050395
19.0 5.758325838e—O5 05528372972 09404648907
20.0 5.282688369e-05 05542289588 0942000332
21.0 4.867535252e—05 0.5555170114 09434259562
22.0 4.502781372e—05 05567132207 09447540867
23.0 4.180382803e—05 05578276006 09459952095
24.0 3.893857494e-05 0558868731 0947158309
25.0 3.63793274e—05 05598440076 0.9482511324
26.0 3.408282432e-05 05607598416 09492803991
27.0 3.201328819e-05 05616218201 09502519686
28.0 3.014091267e-05 05624348355 0.951 1709766
29.0 2.844069649e-05 05632031917 09520419453
30.0 2.68915359e-05 05639306908 09528688739

 

 

 

Table 6.27. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

127

x at y = 0.5 withp= 1 and q = 4 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time $1 $2 $3

0.00 0.1776790146 0.5 0.8223209854
1.00 0006607377161 03792326904 07588257646
2.00 000220046939 04067958452 07706931329
3.00 0001121610614 04347173604 07857362179
4.00 00007402373695 0.4511796931 07984336386
5.00 00005521756939 0461159263 08088016673
6.00 00004366788498 04682746548 08174066602
7.00 00003569970948 04740401834 08247015148
8.00 00002986621627 04790028157 0830998128
9.00 00002544180809 04833734662 08365121318
10.0 0.0002199843118 0.4872611973 08413976884
11.0 00001926105299 04907432172 08457687842
12.0 00001704486633 04938820927 08497120548
13.0 00001522201593 04967295525 08532948084
14.0 00001370190496 04993280139 08565702828
15.0 00001241892226 05017120656 08595812315
16.0 0.000113245834 05039099728 0862362445
17.0 0000103823944 0505945009 08649425679
18.0 9.5644274496—05 05078365278 08673454335
19.0 8.848991239e-05 05096007849 08695910593
20.0 8.219013692e-05 0.5112515593 08716964022
21.0 7.660897502e-05 05128006243 08736759379
22.0 7.163695866e—05 05142581095 08755421114
23.0 6.718509269e-05 05156327856 08773056909
24.0 6.318037653e-05 05169322892 08789760491
25.0 5.956243628e-05 05181633045 08805613887
26.0 5.628096165e-05 05193317105 08820689248
27.0 5.3293734379-05 05204427018 08835050333
28.0 5.056509627e—05 05215008872 08848753732
29.0 4.806474791e-05 05225103727 08861849877
30.0 4.576679829e-05 05234748294 08874383881

 

 

 

Table 6.28. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

128

x at y = 0.5 withp = 1 and q = 1 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time :61 51:2 333

0.00 02502831702 05642793625 08499422355
0.10 01380694362 05448216696 09220406469
0.20 008464881217 05296369026 09516064912
0.30 005599679895 05196505098 09672745255
0.40 003910072152 05131443416 09766478603
0.50 002837399926 05088757108 09827179892
0.60 00211687011 05060494594 0986880513

0.70 001611729371 05041619245 09898584227
0.80 001245896946 05028910896 09920570017
0.90 000974278762 0502028702 0993718743

1.00 0007686885634 05014387613 09949969915
2.00 00008961392785 05000843021 09993932054
3.00 00001169158509 05000095131 09999203926
4.00 1.547189458e—05 05000012352 09999894575
5.00 2.051288952e-O6 05000001633 09999986021
6.00 2.720307154e-07 05000000222 09999998146
7.00 3.607640753e—08 04999999934 09999999754
8.00 4.784437624e-09 05000000303 09999999967
9.00 6.345100001e-10 05000000345 09999999996
10.0 8.415590447e-11 05000064601 09999999999
11.0 1.116406967e-11 05000429575 1

12.0 1.48370205e-12 05002227197 1

13.0 1.962874308e43 04991273235 1

 

 

 

Table 6.29. Gauss-Galerkin Finite Element Method: Changes of the weights at nodes

129

*, + andx at y =05 withp=1 and q: 1 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 1121 (1)2 1123

0.00 008130670793 02760498056 0.1426434865
0.10 009270514293 0.2411981715 01660966856
0.20 01069472399 02031589108 0.1898938493
0.30 01213260558 01691551282 0209518816

0.40 01345030966 01399413413 02255555621
0.50 0146028531 01153124435 02386590255
0.60 01558557559 009477524188 02493690023
0.70 0.1641108891 007776472834 02581243826
0.80 01709822653 006373497565 0265282759

0.90 01766687044 005219544718 02711358484
1.00 01813567149 004272149528 02759217898
2.00 01997940823 0005698205879 0.2945077118
3.00 02022642862 00007561888181 0296979525

4.00 02025922205 00001002939788 02973074855
5.00 02026357167 1.330109301e-05 02973509822
6.00 02026414853 1.763987685e-06 02973567507
7.00 02026422503 2.339393116e-O7 02973575157
8.00 02026423518 3.102492757e-08 02973576172
9.00 02026423652 4.114513505e—09 02973576307
10.0 0202642367 5.456614616e-10 02973576324
11.0 02026423672 7.236511499e-11 02973576327
12.0 02026423673 9.595827 944e-12 0.2973576327
13.0 02026423673 1.272212315e-12 02973576327

 

 

130

Table 6.30. Gauss-Galerkin Finite Element Method: Changes of the “moment”
mj,(y, t) at y = 0.5 with p = 1 and q = 1 as t increases

 

time

m0

m1

m2

m3.

m4

m5

 

0.00

0.5

0.297358

0.196036

0.138456

0.102747

0.0791428

 

0.10

0.5

0.297358

0.214571

0.169451

0.141335

0.122274

 

0.20

0.5

0.297358

0.229715

0.193886

0.17171

0.15665

 

0.30

0.5

0.297358

0.242089

0.213373

0.195746

0.183817

 

0.40

0.5

0.297358

0.252199

0.229037

0.214917

0.2054

 

0.50

 

0.5

0.297358

0.26046

0.241697

0.230316

0.222672

 

0.60

0.5

0.297358

0.267209

0.251966

0.242753

0.23658

 

0.70

0.5

0.297358

0.272724

0.260316

0.252836

0.247831

 

0.80

0.5

0.297358

0.27723

0.267118

0.26103

0.256963

 

0.90

0.5

0.297358

0.280912

0.272663

0.267703

0.264391

 

1.00

0.5

0.297358

0.28392

0.277188

0.273142

0.270443

 

2.00

0.5

0.297358

0.295576

0.294685

0.29415

0.293793

 

3.00

0.5

0.297358

0.297121

0.297003

0.296932

0.296885

 

4.00

0.5

0.297358

0.297326

0.297311

0.297301

0.297295

 

5.00

0.5

0.297358

0.297353

0.297351

0.29735

0.297349

 

6.00

0.5

0.297358

0.297357

0.297357

0.297357

0.297357

 

7.00

0.5

0.297358

0.297358

0.297358

0.297358

0.297357

 

8.00

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

9.00

 

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

10.0

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

11.0

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

12.0

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

13.0

 

 

 

0.5

 

0.297358

 

0.297358

 

0.297358

 

0.297358

 

0.297358

 

 

 

131

Table 6.31. Gauss-Galerkin Finite Element Method: Changes of the “total moment”
M;(t) at y = 0.5 with p = 1 and q = 1 as t increases

 

time

M0

M1

M2

M3

M4

M5

 

0.00

0.5

0.297358

0.196036

0.138456

0.102747

0.0791428

 

0.10

0.5

0.297358

0.214571

0.169451

0.141335

0.122274

 

0.20

0.5

0.297358

0.229715

0.193886

0.17171

0.15665

 

0.30

0.5

0.297358

0.242089

0.213373

0.195746

0.183817

 

0.40

0.5

0.297358

0.252199

0.229037

0.214917

0.2054

 

0.50

0.5

0.297358

0.26046

0.241697

0.230316

0.222672

 

0.60

0.5

0.297358

0.267209

0.251966

0.242753

0.23658

 

0.70

0.5

0.297358

0.272724

0.260316

0.252836

0.247831

 

0.80

0.5

0.297358

0.27723

0.267118

0.26103

0.256963

 

0.90

0.5

0.297358

0.280912

0.272663

0.267703

0.264391

 

1.00

0.5

0.297358

0.28392

0.277188

0.273142

0.270443

 

2.00

0.5

0.297358

0.295576

0.294685

0.29415

0.293793

 

3.00

0.5

0.297358

0.297121

0.297003

0.296932

0.296885

 

4.00

0.5

0.297358

0.297326

0.297311

0.297301

0.297295

 

5.00

0.5

0.297358

0.297353

0.297351

0.29735

0.297349

 

6.00

0.5

0.297358

0.297357

0.297357

0.297357

0.297357

 

7.00

0.5

0.297358

0.297358

0.297358

0.297358

0.297357

 

8.00

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

9.00

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

10.0

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

11.0

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

12.0

0.5

0.297358

0.297358

0.297358

0.297358

0.297358

 

 

13.0

 

 

0.5

 

0.297358

 

0.297358

 

0.297358

 

0.297358

 

0.297358

 

 

132
This agrees with the solution

23411—1 (1 _ myth—1
B(4u. 41/)

 

g(x. y) = (577)

where B(., .) is the beta function. It is concave down.
(3) The solution becomes uniform in y as t increases.

(4) Since :1: = 0 and x = 1 are entrance boundaries, they can not be reached from

the interior. The nodes tend to 0.14647, 0.5 and 0.85355.
(5) The weight for the middle node tends to 0.5 as t increases.

(6) The weights at the other two nodes tend to 0.25 as t increases.

Case II s=0,,u=u=025

In this case, we obtain similar numerical results. The steady state solution agrees
with (6.77) again. It is a straight line.

CmeHIs=Zh=00p=u=0

For this model, both a: = 0 and :1: = 1 are entrance boundaries. The numerical
solution is shown in Figure 6.7, Tables 6.36, 6.37, 6.38 and 6.39. We observe the

following facts:

(1) At t = 30, the solution reaches the steady state based on the tolerance chosen.

(2) Table 6.39 shows that M201) = 1, 114,1,(1) = 1143(1) = Mjf(t) = M36) = Mf,’(t) =
0.330323.

(3) The solutinon becomes uniform in x as t increases.

(4) Since a: = 0 and :1: = 1 are exit boundaries, Table 6.36 shows that the solution
tends to zero at the interior node and “piles” up at x 2 0+ and :1: : 1“ as t

increases.

 

133

(5) The weights for the middle nodes tend to zero as t increases.

(6) The weights at the :1: = 0 is 0.169677 and 0.830322 at :1: : 1as t increases.

134

 

 

 

 

1 xxxxixxxfixxxxxele"“““rnf‘T
xx
x
x
x d
0.9,}-
O.8'- -‘
0.71- .
x061' -1
.92
a
.9
go.5++++++++++++++++++++++++++++++<r
3
20.4- -
0.35 4
0.2:- -
“1 .-
0.1-*.
’.**‘
"!§.
0 W4
0 0.5 1 1.5

Tnme Variable t

Figure 6.1. Gauss-Galerkin Finite Element Method: Movement of the nodes 1:, +
and x as t increases

135

 

 

 

”0'0ooooooooooooooooooooooooooooia
0.7- .

X0.6" d

2

.D

g

gosiL-xxxxxxxxxxxxxxxxxxxxxxxxxxxxx-u

O)

‘O

o

20.4— «
0.35 .
021‘“++~~I~++++++++++++++++++++++++;-
0.1.. Cl
.-
0W
0 0.5 1 1.5

Time Variable!

Figure 6.2. Gauss-Galerkin Finite Element Method: Movement of the nodes 0, o, x,
+ and :1: as t increases

136

 

0.8 -

Node Variable x

 

X

+++++++++++++++++++++++++++++

T

)(x"
x"

01

*1:
*fifuateagaqy.

10 15
Time Variable!

xxixxxxIXXXXXYXXTxxxxx

la

 

{gigggggggggr
20 25 30

Figure 6.3. Gauss-Galerkin Finite Element Method: Movement of the nodes *, +
and x with p = 2 and q = 2 as t increases

Node Variable x

137

 

 

 

1 :‘x x x x i x x x x X X X x x t x x x w x x x x x
x
X
x
O.9~ " -'
i:
0.8- .1
0.7- a
05' +++++++++++++1»
+ +
+ + +
0.5- + -
0.4- «
0-3- q
0.2- ..
0.1- -
oW—l—i—i—i—I—u—I—l—é—H—i—i—t—e—u—u—H—u—W
O 5 10 15 20 25 30

Time Variable!

Figure 6.4. Gauss-Galerkin Finite Element Method: Movement of the nodes at, +
and x with p = 1 and q = 2 as t increases

138

 

0.9 .

+
.1.

+
+++.1.

 

Node Variable x
O
U!
l

+++++++++++++++++++++y

'1
-4

411*.
1****?*§§gg£“glggig...-.L

5 1o 15 20 25 30
Time Variable!

 

Figure 6.5. Gauss-Galerkin Finite Element Method: Movement of the nodes *, +
and x with p = 2 and q = 1 as t increases

139

 

 

 

 

1 1 l I 1 T
0.9- ‘
xxxxXXXXXXXXXXXXXXXXXXXXXXXXIK
11" ..
08"
o.7— ‘
31:06p ‘
2
8
§o_5.-+++++++++++++++++++++++++++++~~
§
20.4- ‘
03. 1
0.2r ‘
T“*eeeeaeaeeeeeeaaeeeeeeeeeieefi
o.1~ ‘
O 1 1 l l J
O 1 2 3 4 5 6

Time Variable!

Figure 6.6. Gauss-Galerkin Finite Element Method: Movement of the nodes *, +
and x at y = 0.5 with s = 0,11 = 0375,11 = 0.375 as t increases

140

 

 

 

 

 

X
0.91" "
I!
08" '1
0.7" .4
30-5"+++++++++++++++++++++++++++++1
3
.g ‘
>05»
Q3
8
20.4»- ‘
0.3) ..
02" _
I'
01- -
4'
.1
0 1' *J—Las—a—e—t—H—u—H—a—n—H—t—H—e—I—WH
0 5 10 15 20 25 30

Time Variable!

Figure 6.7. Gauss-Galerkin Finite Element Method: Movement of the nodes :11, +
and x at y = 0.5 with s = 2,h= 05,11: 0,1/ = 0 as t increases

141

Table 6.32. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and
x at y = 0.5 with s = 0, [.1 = 0375,11 = 0.375 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 221 3:2 273

0.0 0.1776790146 0.5 0.8223209854
0.2 01642962137 0.5 08357037863
0.4 01574239668 0.5 08425760332
0.6 01534135247 0.5 08465864753
0.8 01509533372 0.5 08490466628
1.0 0.1493957345 0.5 08506042655
1.2 01483902464 0.5 08516097536
1.4 0147733336 0.5 0852266664

1.6 01473009068 0.5 08526990932
1.8 01470148679 0.5 08529851321
2.0 01468250667 0.5 08531749333
2.2 01466988646 0.5 0.8533011354
2.4 01466148369 0.5 08533851631
2.6 01465588394 0.5 08534411606
2.8 01465214994 0.5 08534785006
3.0 01464965907 0.5 08535034093
3.2 01464799703 0.5 08535200297
3.4 01464688782 0.5 08535311218
3.6 01464614749 0.5 08535385251
3.8 01464565331 0.5 08535434669
4.0 01464532343 0.5 08535467657
4.2 01464510321 0.5 08535489679
4.4 0146449562 0.5 0853550438

4.6 01464485805 0.5 08535514195
4.8 01464479253 0.5 08535520747
5.0 01464474879 0.5 08535525121
5.2 01464471959 0.5 08535528041
5.4 0146447001 0.5 0853552999

5.6 01464468708 0.5 08535531292
5.8 01464467839 0.5 08535532161
6.0 01464467259 0.5 08535532741

 

 

 

 

 

 

142

Table 6.33. Gauss-Galerkin Finite Element Method: Changes of the weights at nodes
*, + and x at y = 0.5 with s = 0,11 = 0375,12 = 0.375 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 1121 1122 1.113

0.0 0.2279202041 0.5441595918 0.2279202041
0.2 02325481017 05349037966 0.2325481017
0.4 02375344137 0.5249311726 0.2375344137
0.6 02413979521 05172040959 0.2413979521
0.8 02441527197 05116945607 0.2441527197
1.0 02460550664 05078898671 02460550664
1.2 02473494077 0.5053011846 0.2473494077
1.4 02482233103 05035533794 02482233103
1.6 02488108053 05023783893 0.2488108053
1.8 02492047529 05015904942 0.2492047529
2.0 02494685031 05010629938 0.2494685031
2.2 0.2496449112 05007101777 0.2496449112
2.4 0249762826 0500474348 0.249762826
2.6 02498416103 05003167795 0.2498416103
2.8 02498942353 0.5002115294 0.2498942353
3.0 02499293807 05001412386 0.2499293807
3.2 02499528495 05000943009 0.2499528495
3.4 024996852 050006296 0.24996852
3.6 02499789828 05000420344 0.2499789828
3.8 02499859684 05000280633 0.2499859684
4.0 02499906322 05000187356 0.2499906322
4.2 02499937459 05000125082 0.2499937459
4.4 02499958247 05000083506 0.2499958247
4.6 02499972125 0500005575 0.2499972125
4.8 0249998139 05000037219 0.249998139
5.0 02499987576 05000024848 0.2499987576
5.2 02499991706 05000016589 02499991706
5.4 02499994463 05000011075 02499994463
5.6 02499996303 05000007394 02499996303
5.8 02499997532 05000004936 02499997532
6.0 02499998352 05000003295 02499998352

 

 

 

 

 

 

143

Table 6.34. Gauss-Galerkin Finite Element Method: Changes of the “moment”
m;(y, t) at y = 0.5 with s = 0,11 = 0.375, V = 0.375 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time mo m1 m2 m3 m4 mg,

0.0 1 0.5 0.297358 0.196036 0.138456 0.102747
0.2 1 0.5 0.302391 0.203586 0.146988 0.111492
0.4 1 0.5 0.305751 0.208627 0.152669 0.117295
0.6 1 0.5 0.307994 0.211992 0.156458 0.121159
0.8 1 0.5 0.309492 0.214238 0.158986 0.123735
1.0 1 0.5 0.310492 0.215738 0.160674 0.125454
1.2 1 0.5 0.311159 0.216739 0.1618 0.126602
1.4 1 0.5 0.311605 0.217407 0.162552 0.127368
1.6 1 0.5 0.311902 0.217854 0.163054 0.127879
1.8 1 0.5 0.312101 0.218152 0.163389 0.128221
2.0 1 0.5 0.312234 0.218351 0.163613 0.128449
2.2 1 0.5 0.312322 0.218483 0.163762 0.128601
2.4 1 0.5 0.312381 0.218572 0.163862 0.128702
2.6 l 0.5 0.312421 0.218631 0.163929 0.12877

2.8 1 0.5 0.312447 0.218671 0.163973 0.128815
3.0 1 0.5 0.312465 0.218697 0.164003 0.128846
3.2 1 0.5 0.312476 0.218715 0.164023 0.128866
3.4 1 0.5 0.312484 0.218726 0.164036 0.128879
3.6 1 0.5 0.312489 0.218734 0.164045 0.128888
3.8 1 0.5 0.312493 0.218739 0.164051 0.128894
4.0 1 0.5 0.312495 0.218743 0.164055 0.128898
4.2 1 0.5 0.312497 0.218745 0.164057 0.128901
4.4 1 0.5 0.312498 0.218747 0.164059 0.128903
4.6 1 0.5 0.312499 0.218748 0.16406 0.128904
4.8 1 0.5 0.312499 0.218749 0.164061 0.128905
5.0 1 0.5 0.312499 0.218749 0.164061 0.128905
5.2 1 0.5 0.3125 0.218749 0.164062 0.128906
5.4 1 0.5 0.3125 0.21875 0.164062 0.128906
5.6 1 0.5 0.3125 0.21875 0.164062 0.128906
5.8 1 0.5 0.3125 0.21875 0.164062 0.128906
6.0 1 0.5 0.3125 0.21875 0.164062 0.128906

 

 

 

144

Table 6.35. Gauss-Galerkin Finite Element Method: Changes of the “total moment”
M,‘,(t) at y=05 with 3 = 0, p = 0.375, V = 0.375 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5
0.0 1 0.5 0.297358 0.196036 0.138456 0.102747
0.2 1 0.5 0.302391 0.203586 0.146988 0.111492
0.4 1 0.5 0.305751 0.208627 0.152669 0.117295
0.6 1 0.5 0.307994 0.211992 0.156458 0.121159
0.8 1 0.5 0.309492 0.214238 0.158986 0.123735
1.0 1 0.5 0.310492 0.215738 0.160674 0.125454
1.2 1 0.5 0.311159 0.216739 0.1618 0.126602
1.4 1 0.5 0.311605 0.217407 0.162552 0.127368
1.6 1 0.5 0.311902 0.217854 0.163054 0.127879
1.8 1 0.5 0.312101 0.218152 0.163389 0.128221
2.0 1 0.5 0.312234 0.218351 0.163613 0.128449
2.2 1 0.5 0.312322 0.218483 0.163762 0.128601
2.4 1 0.5 0.312381 0.218572 0.163862 0.128702
2.6 1 0.5 0.312421 0.218631 0.163929 0.12877
2.8 1 0.5 0.312447 0.218671 0.163973 0.128815
3.0 1 0.5 0.312465 0.218697 0.164003 0.128846
3.2 1 0.5 0.312476 0.218715 0.164023 0.128866
3.4 1 0.5 0.312484 0.218726 0.164036 0.128879
3.6 1 0.5 0.312489 0.218734 0.164045 0.128888
3.8 1 0.5 0.312493 0.218739 0.164051 0.128894
4.0 1 0.5 0.312495 0.218743 0.164055 0.128898
4.2 1 0.5 0.312497 0.218745 0.164057 0.128901
4.4 1 0.5 0.312498 0.218747 0.164059 0.128903
4.6 1 0.5 0.312499 0.218748 0.16406 0.128904
4.8 1 0.5 0.312499 0.218749 0.164061 0.128905
5.0 l 0.5 0.312499 0.218749 0.164061 0.128905
5.2 1 0.5 0.3125 0.218749 0.164062 0.128906
5.4 1 0.5 0.3125 0.21875 0.164062 0.128906
5.6 1 0.5 0.3125 0.21875 0.164062 0.128906
5.8 1 0.5 0.3125 0.21875 0.164062 0.128906
6.0 1 0.5 0.3125 0.21875 0.164062 0.128906

 

 

 

 

 

 

 

 

 

Table 6.36. Gauss-Galerkin Finite Element Method: Changes of the nodes *, + and

145

xaty=05 withs=2,h=05,,u=0,V=0ast increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time 3:1 :132 1);;

0.00 0.1776790146 0.5 0.8223209854
1.00 007823484538 05855926881 09672094934
2.00 004055785446 05952942733 09882813902
3.00 002175701743 05935687484 0994844609

4.00 0.01176627035 05905610497 09975060185
5.00 0006369293956 05882085524 09987326259
6.00 0003445438578 0.5866681196 09993384161
7.00 0001862301052 058573549 0999649455

8.00 0001006005904 05851946307 09998127142
9.00 00005432371344 05848889872 09998994783
10.0 00002932798893 05847190463 09999459106
11.0 00001583142674 05846255329 09999708549
12.0 8.54529272e-O5 05845744192 09999842838
13.0 4.612291437e-05 05845466024 09999915218
14.0 2.489414189e—05 05845315072 09999954253
15.0 1.343607637e-05 0584523331 09999975313
16.0 7.251786085e-06 05845189076 09999986677
17.0 3.913956526e-06 05845165166 0999999281

18.0 2.112448808e-06 05845152244 0.9999996119
19.0 1.140134147e-06 05845145278 09999997906
20.0 6.153546255e—07 05845141503 0999999887

21.0 3.321198762e-07 05845139491 0999999939

22.0 1.792520631e-07 05845138348 09999999671
23.0 9.674608925e—08 05845137629 09999999822
24.0 5.221588395e-08 05845137595 09999999904
25.0 2.81820044e—08 05845137242 09999999948
26.0 1 .521042681e-08 05845137523 09999999972
27.0 . 8.209376512e09 05845136238 09999999985
28.0 4.430779077e—09 05845135372 09999999992
29.0 2.391380427e-09 05845133642 09999999996
30.0 1.290684337e-09 05845139703 09999999998

 

 

 

 

Table 6.37. Gauss—Galerkin Finite Element Method: Changes of the weights at nodes

146

*, + andxaty=05withs=2,h=05,;1=0,V=0astincreases

 

time

011

W2

W3

 

0.00

0.2279202041

0.5441595918

0.2279202041

 

1.00

0203454735

03434686623

04530766027

 

2.00

01877795851

0.1832292954

06289911195

 

3.00

01795033149

009603249769

0.7244641874

 

4.00

01750216309

005062793336

07743504358

 

5.00

01725788113

002690733567

0800513853

 

6.00

0.1712492829

001438787971

08143628374

 

7.00

01705276659

0007723698604

08217486355

 

8.00

0.1701368564

0004155979967

08257071636

 

9.00

0169925507

0002239281609

08278352114

 

10.0

01698113081

0001207464186

08289812277

 

11.0

0.1697496338

00006513627194

08295990035

 

12.0

01697163352

00003514569213

08299322079

 

13.0

0.1696983599

00001896601995

08301119799

 

14.0

0.1696886572

00001023552587

08302089876

 

15.0

0.1696834201

5.524084473e-05

0830261339

 

16.0

0.1696805935

2.981393018e—05

08302895925

 

17.0

0.1696790679

1.6090994716-05

0.8303048411

 

18.0

0.1696782445

8.684585931e—06

08303130709

 

19.0

01696778001

4.687234992e-06

08303175127

 

20.0

0.1696775602

2.52979357e-06

083031991

 

21.0

0.1696774308

1.3653809746—06

08303212039

 

22.0

0.1696773609

7.369242155e-07

08303219022

 

23.0

0.1696773232

3.977332705e-07

08303222791

 

24.0

01696773028

2.146649293e—07

08303224825

 

25.0

0.1696772918

1.158591251e-07

08303225923

 

26.0

0.1696772859

6.253159069e-08

08303226516

 

27.0

0.1696772827

3.374960315e-08

08303226835

 

28.0

0.169677281

1.8215360866-08

08303227008

 

29.0

0.1696772801

9.831214716e-09

08303227101

 

 

30.0

 

 

0.1696772795

 

5.3061115626-09

 

08303227151

 

 

147

Table 6.38. Gauss-Galerkin Finite Element Method: Changes of the “moment”
m;(y, t) at y = 0.5 with s = 2, h = 05,11 = 0, V = 0 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time m0 m1 m2 m3 m4 m5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
1.00 1 0.65527 0.542878 0.479022 0.436907 0.40716
2.00 1 0.738312 0.679577 0.645803 0.623033 0.606688
3.00 1 0.781637 0.750933 0.733402 0.72156 0.713057
4.00 1 0.804377 0.788174 0.778999 0.772813 0.768379
5.00 1 0.816426 0.807803 0.80295 0.799684 0.797349
6.00 1 0.822855 0.81824 0.815653 0.813914 0.812672
7.00 1 0.826302 0.823823 0.822437 0.821506 0.820842
8.00 1 0.828156 0.826821 0.826076 0.825576 0.825219
9.00 1 0.829154 0.828435 0.828034 0.827764 0.827572
10.0 1 0.829692 0.829304 0.829088 0.828943 0.82884
11.0 1 0.829983 0.829773 0.829657 0.829578 0.829523
12.0 1 0.830139 0.830026 0.829963 0.829921 0.829891
13.0 1 0.830224 0.830163 0.830129 0.830106 0.83009
14.0 1 0.830269 0.830236 0.830218 0.830206 0.830197
15.0 1 0.830294 0.830276 0.830266 0.83026 0.830255
16.0 1 0.830307 0.830298 0.830292 0.830289 0.830286
17.0 1 0.830314 0.830309 0.830306 0.830304 0.830303
18.0 1 0.830318 0.830315 0.830314 0.830313 0.830312
19.0 1 0.83032 0.830319 0.830318 0.830317 0.830317
20.0 1 0.830321 0.830321 0.83032 0.83032 0.83032
21.0 1 0.830322 0.830322 0.830321 0.830321 0.830321
22.0 1 0.830322 0.830322 0.830322 0.830322 0.830322
23.0 1 0.830323 0.830322 0.830322 0.830322 0.830322
24.0 1 0.830323 0.830323 0.830323 0.830322 0.830322
25.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
26.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
27.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
28.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
29.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
30.0 1 0.830323 0.830323 0.830323 0.830323 0.830323

 

 

Table 6.39. Gauss-Galerkin Finite Element Method: Changes of the “total moment”

148

Mf,(t) at y=05 with s = 2, h = 05,11 = 0, V = 0 as t increases

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time M0 M1 M2 M3 M4 M5
0.00 1 0.5 0.297358 0.196036 0.138456 0.102747
1.00 1 0.65527 0.542878 0.479022 0.436907 0.40716
2.00 1 0.738312 0.679577 0.645803 0.623033 0.606688
3.00 1 0.781637 0.750933 0.733402 0.72156 0.713057
4.00 1 0.804377 0.788174 0.778999 0.772813 0.768379
5.00 1 0.816426 0.807803 0.80295 0.799684 0.797349
6.00 1 0.822855 0.81824 0.815653 0.813914 0.812672
7.00 1 0.826302 0.823823 0.822437 0.821506 0.820842
8.00 1 0.828156 0.826821 0.826076 0.825576 0.825219
9.00 1 0.829154 0.828435 0.828034 0.827764 0.827572
10.0 1 0.829692 0.829304 0.829088 0.828943 0.82884
11.0 1 0.829983 0.829773 0.829657 0.829578 0.829523
12.0 1 0.830139 0.830026 0.829963 0.829921 0.829891
13.0 1 0.830224 0.830163 0.830129 0.830106 0.83009
14.0 1 0.830269 0.830236 0.830218 0.830206 0.830197
15.0 1 0.830294 0.830276 0.830266 0.83026 0.830255
16.0 1 0.830307 0.830298 0.830292 0.830289 0.830286
17.0 1 0.830314 0.830309 0.830306 0.830304 0.830303
18.0 1 0.830318 0.830315 0.830314 0.830313 0.830312
19.0 1 0.83032 0.830319 0.830318 0.830317 0.830317
20.0 1 0.830321 0.830321 0.83032 0.83032 0.83032
21.0 1 0.830322 0.830322 0.830321 0.830321 0.830321
22.0 1 0.830322 0.830322 0.830322 0.830322 0.830322
23.0 1 0.830323 0.830322 0.830322 0.830322 0.830322
24.0 1 0.830323 0.830323 0.830323 0.830322 0.830322
25.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
26.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
27.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
28.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
29.0 1 0.830323 0.830323 0.830323 0.830323 0.830323
30.0 1 0.830323 0.830323 0.830323 0.830323 0.830323

 

 

 

 

CHAPTER 7

Conclusions and Discussions

In the previous chapters, we proposed and studied a Gauss-Galerkin finite element
method for solving a class of singular diffusion equations. The theoretic analysis is
very general and therefore the results can be applied to a wide range of singular diffu-
sion equations in two variables. In our examination of the test problems, even though
we only use very few base functions in finite element approximation for y variable
and very few nodes in Gauss-Galerkin approximation for x variable, the numerical
approximation seems very efficient and accurate. In the proof of the convergence
of Gauss-Galerkin finite method, we considered a set of boundary conditions under
which the boudary terms drop out. Nevertheless, as our test problem in studing
Section 6.2 shows, we can apply the proposed method even in those case where the
boundary terms do not all drop out (there are net fluxes across the boundaries at
:1: = 0 and x = 1).

A number of important and interesting issues remain to be studied in the future. We
state some of them here. How do we modify our Gauss-Galerkin method in general
in the case that the boundary terms can not drop out. One possible solution is to
approximate such boundary terms by the finite element method in the y variable
and the Gauss-Galerkin method in the x variable. For singular diffusion equations

with singularities in both x and y variables, a possible solution may need the Gauss-

149

 

150

Galerkin method in both directions. For example, given time t, we may alternately
use one dimensional Gauss-Galerkin method in the x direction and then the y direc-

tion repeatedly. Obviously much more work needs to be done in the future.

 

 

BIBLIOGRAPHY

BIBLIOGRAPHY

[1] D. A. Dawson, Galerkin approximation of nonlinear Markov processes, Statistics
and Related Topics (Proceeding of the International Symposium on Statistics
and Related Topics held in Ottawa, Canada, May 5-7, 1980), North Holland,
317-339, 1981

[2] W. Feller, The parabolic differential equations and the associated semi-groups of
transformations, Ann. Math., 55, 468-519, 1952.

[3] Friedman, Stochastic Differential Equations and Application to P.D.E. ’3, Vol. 1,
Academic Press, Inc. 1975

[4] A. Hajjafar, H. Salehi, D. H. Y. Yen, On a class of Gauss-Galerkin methods for
Markov Processes, Proceeding of the Workshop on Nonstationary Stochastic Pro-
cesses and Their Applications, August 1-2, 1991, pp.126—146, World Scientific,
Virginia 1992.

[5] L. Huang and D. H. Y. Yen, A Gauss-Galerkin Finite-Difference Method for Sin-
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