Adaptive space-time BEM for the heat equation

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.