The electromagnetic scattering from an open large cavity embedded in an infinite plane is of practical importance due to its significant industrial and military applications. Examples of cavities include jet engine inlet ducts, exhaust nozzles and cavity-backed antennas [3, 4]. In many practical applications, one is interested in the cavity problem with either a large wave number $k$ or a large diameter $a$, in which case the solution has a highly oscillatory nature [7]. While the original time-harmonic problem is modeled by Helmholtz equation in the unbounded domain, the reformulated model through Fourier transform is essentially a Helmholtz equation in the bounded domain with mixed nonlocal boundary condition. Deriving an explicit dependency between the wave energy and the wave number is mathematically interesting and challenging. The stability estimate is also important as it defines relations between the wave number and the discretization parameters in the error analysis [16]. For the open cavity problem, while the stability analysis for the rectangular cavity was derived recently [8] as described , the stability results for more general shapes of cavities are to be explored. The objective of this thesis work is to partially answer this question by imposing some geometric assumptions.We first start from considering a class of cavity with a strong geometric constraint. The energy stability is established by careful choices of the parameters, and test functions, which take full advantage of geometric properties. The arguments are based on the appropriate usage of the real and imaginary part of the weak formulation of the problem, the separation of lower frequency and higher frequency part, and connections between frequency components and spatial components. The energy in cavity is bounded by the energy of incoming field with coefficient in terms of powers of wave number. Next, we investigate the case where a weaker geometric constraint is imposed. A new auxiliary function with compact support near the boundary of the cavity is carefully constructed to reformulate the problem. However, the original homogeneous Helmholtz equation is changed to a non-homogeneous one, all previous work in homogeneous equation must be suitably modified, and the estimate in terms of wave number $k$ is obtained from detailed analysis of this auxiliary function. The energy norm is proved to be at most in the order of $k^{frac{7}{10}}$, which is the same in terms of the power of wave number $k$ as the case with strong geometric conditions but with other additional terms. Furthermore, we studied the case where the cavity domain is of rectangular-like shape, where new test function is introduced and new inequalities are established to derive the energy estimate.
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