Adaptive space-time BEM for the heat equation

TL;DR

This paper develops an adaptive space-time boundary element method for the heat equation, introducing a residual error estimator that effectively guides mesh refinement and achieves optimal convergence rates.

Contribution

It proposes a novel residual-type a posteriori error estimator for space-time BEM, ensuring efficiency, reliability, and optimal adaptive refinement for the heat equation.

Findings

Estimator is both a lower and an upper bound for the BEM error.

Adaptive anisotropic refinement converges at the best possible rate.

Estimator performs well in numerical experiments on 2D spatial domains.

Abstract

We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted $L_2$-norms of the residual, also an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$. While the results do not depend on the local aspect ratio between time and space, assuming the scaling $|J| \eqsim {\rm diam}(K)^2$ for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional $L_2$-terms. In the considered numerical experiments on two-dimensional domains in space, the estimator seems to be equivalent to the error, independently of these assumptions. In particular for adaptive anisotropic refinement, both converge with the best possible convergence rate.