This thesis focuses on two related topics, which are to design efficient discontinuous Galerkin (DG) schemes for Hamilton-Jacobi (HJ) equations and high-dimensional elliptic equations.In the first part, we propose a new DG method that solves for the viscosity solution of the general HJ equations. The new method is compact and easy to implement. We avoid the reconstruction of the solution across elements by utilizing the interfacial terms involving the Roe speed. A penalty term proportional to the jump of the normal derivative of the numerical solution is added to fix the entropy violation, which was inspired by the Harten and Hymans entropy fix [53] for Roe scheme for the conservation laws. Numerical experiments demonstrate good performance for general Hamiltonians, including nonconvex Hamiltonians.In the second part, we develop an interior penalty DG method on sparse grids for efficient computations of high-dimensional second-order elliptic problems. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degree of freedom from the standard O(h−d) to O(h−1|log2h|d−1) for d-dimensional problems, where h is the uniform mesh size in each dimension. Compared to the traditional full grid approaches, the accuracy of the numerical approximation of this method is only slightly deteriorated by a factor of | log2 h|d−1 in the energy norm. Error estimates are provided and confirmed by numerical tests in multi-dimensions.
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