Sparse Grid Approximation Spaces for Space-Time Boundary Integral Formulations of the Heat Equation

TL;DR

This paper introduces a novel sparse grid discretisation approach for boundary integral formulations of the heat equation, achieving higher convergence rates and improved accuracy over traditional methods.

Contribution

It develops a-priori error analysis for sparse grid discretisations, demonstrating their superior convergence and proposing an adaptive scheme for optimal grid shape.

Findings

Sparse grid discretisation yields higher convergence rates.

Improved accuracy for low polynomial degree discretisations.

Adaptive scheme suggests optimal sparse grid index sets.

Abstract

The aim of this paper is to develop stable and accurate numerical schemes for boundary integral formulations of the heat equation with Dirichlet boundary conditions. The accuracy of Galerkin discretisations for the resulting boundary integral formulations depends mainly on the choice of discretisation space. We develop a-priori error analysis utilising a proof technique that involves norm equivalences in hierarchical wavelet subspace decompositions. We apply this to a full tensor product discretisation, showing improvements over existing results, particularly for discretisation spaces having low polynomial degrees. We then use the norm equivalences to show that an anisotropic sparse grid discretisation yields even higher convergence rates. Finally, a simple adaptive scheme is proposed to suggest an optimal shape for the sparse grid index sets.