Semi-analytic integration for a parallel space-time boundary element method modeling the heat equation

Abstract

The presented paper concentrates on the boundary element method (BEM) for the heat equation in three spatial dimensions. In particular, we deal with tensor product space-time meshes allowing for quadrature schemes analytic in time and numerical in space. The spatial integrals can be treated by standard BEM techniques known from three dimensional stationary problems. The contribution of the paper is twofold. First, we provide temporal antiderivatives of the heat kernel necessary for the assembly of BEM matrices and the evaluation of the representation formula. Secondly, the presented approach has been implemented in a publicly available library besthea allowing researchers to reuse the formulae and BEM routines straightaway. The results are validated by numerical experiments in an HPC environment.

An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions

“…The second term in this representation is similar to what we have formally derived in (46), so Theorem 5.2 justifies the calculations found in the literature. In [24] it is described in detail how to deal with the remaining integrals. We conclude this section with a proof of Theorem 5.2.…”

A parallel fast multipole method for a space-time boundary element method for the heat equation

“…M h denotes the related mass matrix. Please check [25] for details on the discretization and implementation. Some detailed analysis of the integral equations and the presented boundary element method is provided in [5,26].…”

“…for all (k t , k x ) ∈ Ẑtar , where w jt,jx denotes a coefficient of the vector w. The computation of the corresponding integrals is discussed in [25].…”

Adaptive space-time BEM for the heat equation

“…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, Tau19,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.…”

“…Two often mentioned advantages of simultaneous space-time methods are their potential for massive parallelization as well as their potential for fully adaptive refinement to resolve singularities local in both space and time. While the first advantage has been investigated in, e.g., [DZO + 19,ZWOM21], the latter requires suitable a posteriori computable error estimators, which have not been developed yet for the heat equation. Indeed, concerning a posteriori error estimation as well as adaptive refinement for BEM for time-dependent problems, we are only aware of the works [Glä12, G ÖSS20] for the wave equation in two and three space dimensions, respectively.…”