Adaptive space-time BEM for the heat equation

Gregor Gantner,Raymond Van Venetië

doi:10.1016/j.camwa.2021.12.022

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 L2-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×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|≂diam(K)2 for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional L2-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.

Highlights

In the last years, there has been a growing interest in simultaneous space-time boundary element methods (BEM) for the heat equation [CS13, MST14, MST15, HT18, CR19, DNS19, DZO+19, Tau[19], ZWOM21]
In contrast to the differential operator based variational formulation on the space-time cylinder, the variational formulation corresponding to space-time BEM is coercive [AN87, Cos90] so that the discretized version always has a unique solution regardless of the chosen trial space which is even quasi-optimal in the natural energy norm
The potential disadvantage that discretizations lead to dense matrices due to the nonlocality of the boundary integral operators has been tackled, e.g., in [MST14, MST15, HT18] via the fast multipole method and H-matrices

Summary

Introduction

There has been a growing interest in simultaneous space-time boundary element methods (BEM) for the heat equation [CS13, MST14, MST15, HT18, CR19, DNS19, DZO+19, Tau[19], ZWOM21]. In contrast to the differential operator based variational formulation on the space-time cylinder, the variational formulation corresponding to space-time BEM is coercive [AN87, Cos90] so that the discretized version always has a unique solution regardless of the chosen trial space which is even quasi-optimal in the natural energy norm It is naturally applicable on unbounded domains and only requires a mesh of the lateral boundary of the space-time cylinder resulting in a dimension reduction. Space-time boundary element method, heat equation, a posteriori error estimation, adaptive mesh-refinement, computation of singular integrals. The remainder of this work is organized as follows: Section 2 summarizes the general principles of the space-time boundary element method for the heat equation. This result is invoked in Corollary 3.5 for the residual, resulting in efficient and reliable a posteriori computable error bounds.

Preliminaries

A posteriori error estimation

Numerical experiments