Efficient Ritz-Galerkin Discretization of PDEs with Variable Coefficients in arbitrary Dimensions using Sparse Grids

Sparse grids can be used to discretize elliptic differential equations of second order on a d-dimensional cube. Using the Ritz-Galerkin discretization, one obtains a linear equation system with 𝒪 (N (log N)^d−1) unknowns. The corresponding discretization error is 𝒪 (N⁻¹ (log N)^d−1) in the H¹-norm. A major difficulty in using this sparse grid discretization is the complexity of the related stiffness matrix. To reduce the complexity of the sparse grid discretization matrix, we apply prewavelets and a discretization with semi-orthogonality. Furthermore, a recursive algorithm is used, which performs a matrix vector multiplication with the stiffness matrix by 𝒪 (N (log N)^d−1) operations. Simulation results up to level 10 are presented for a 6-dimensional Helmholtz problem with variable coefficients. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)