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

Abstract

AbstractSparse 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)