Space-time boundary element methods for the heat equation

In this chapter, we describe a space-time boundary element method for the numerical solution of the time-dependent heat equation. As model problem, we consider the initial Dirichlet boundary value problem, where the solution can be expressed in terms of given Dirichlet and initial data, and the unknown Neumann datum, which is determined by the solution of an appropriate boundary integral equation. For its numerical approximation, we consider a discretization, which is done with respect to a space-time decomposition of the boundary of the space-time domain. This space-time discretization technique allows us to parallelize the computation of the global solution of the whole space-time system. Besides the widely-used tensor product approach, we also consider an arbitrary decomposition of the space-time boundary into boundary elements, allowing us to apply adaptive refinement in space and time simultaneously. In addition to the analysis of the boundary integral operators and the formulation of boundary element methods for the initial Dirichlet boundary value problem, we state a priori error estimates of the approximations. Moreover, we present numerical experiments to confirm the theoretical findings.