Sparse Grid Discontinuous Galerkin Methods for High-Dimensional Elliptic Equations

TL;DR

This paper introduces sparse grid discontinuous Galerkin methods for high-dimensional elliptic PDEs, significantly reducing computational complexity and overcoming the curse of dimensionality while maintaining accuracy.

Contribution

The paper develops a novel sparse grid DG framework using hierarchical bases, reducing degrees of freedom from exponential to nearly linear in dimension, and provides theoretical error estimates.

Findings

Achieves accuracy of O(h^k |log h|^{d-1}) in energy norm.

Reduces degrees of freedom from O(h^{-d}) to O(h^{-1} |log h|^{d-1}).

Numerical tests confirm theoretical error estimates.

Abstract

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many applications due to their distinctive features. However, they are often deemed too costly because of the large number of degrees of freedom of the approximation space, which are the main bottleneck for simulations in high dimensions. In this paper, we develop sparse grid DG methods for elliptic equations with the aim of breaking the \emph{curse of dimensionality}. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degrees of freedom from the standard {$O(h^{-d})$ to $O(h^{-1}|\log_2 h|^{d-1})$} for $d$-dimensional problems, where $h$ is the uniform mesh size in each dimension. Our method, based on the interior penalty (IP) DG framework, can achieve accuracy of $O(h^{k}|\log_2 h|^{d-1})$ in the energy norm, where $k$ is the degree of polynomials used. Error estimates are provided and confirmed by numerical tests in multi-dimensions.