An introduction to the analysis and implementation of sparse grid finite element methods

TL;DR

This paper introduces an accessible approach to analyzing and implementing sparse grid finite element methods, highlighting their efficiency in solving PDEs with fewer degrees of freedom and providing MATLAB code for practical application.

Contribution

It presents an elementary analysis and implementation of sparse grid finite element methods, including a multiscale extension, with MATLAB code and cost-accuracy comparisons.

Findings

Sparse grid methods achieve accurate PDE solutions with fewer degrees of freedom.

The multiscale sparse grid method extends the two-scale approach effectively.

Comparative analysis shows efficiency gains over classical finite element methods.

Abstract

Our goal is to present an elementary approach to the analysis and programming of sparse grid finite element methods. This family of schemes can compute accurate solutions to partial differential equations, but using far fewer degrees of freedom than their classical counterparts. After a brief discussion of the classical Galerkin finite element method with bilinear elements, we give a short analysis of what is probably the simplest sparse grid method: the two-scale technique of Lin et al. (2001). We then demonstrate how to extend this to a multiscale sparse grid method which, up to choice of basis, is equivalent to the hierarchical approach, as described by, e.g., Bungartz and Griebel (2004). However, by presenting it as an extension of the two-scale method, we can give an elementary treatment of its analysis and implementation. For each method considered, we provide MATLAB code, and a comparison of accuracy and computational costs.