A Sparse Grid Discretization with Variable Coefficient in High Dimensions

TL;DR

This paper introduces a sparse grid Ritz-Galerkin discretization method with pre-wavelets for solving high-dimensional elliptic PDEs with variable coefficients, demonstrating effective convergence and manageable condition numbers.

Contribution

The paper develops a new sparse grid discretization approach using pre-wavelets for high-dimensional elliptic equations with variable coefficients, including an efficient matrix-vector multiplication algorithm.

Findings

Convergence aligns with finite element approximation properties.

Condition number of the stiffness matrix is bounded below 10.

Method successfully applied to 3D and 6D problems with variable coefficients.

Abstract

We present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension $d=2,3$ and higher dimensions $d>3$. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple pre-wavelet stencil, and the classical operator dependent stencil for multilinear finite elements. Numerical simulation results are presented for a 3-dimensional problem on a curvilinear bounded domain and for a 6-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below $10$ using a standard diagonal preconditioner.