Time-dependent Steklov--Poincaré operators and space-time Robin--Robin decomposition for the heat equation

Abstract: Domain decomposition methods are a set of widely used tools for parallelization of partial differential equation solvers. Convergence is well studied for elliptic equations, but in the case of parabolic equations there are hardly any results for general Lipschitz domains in two or more dimensions. The aim of this work is therefore to construct a new framework for analyzing nonoverlapping domain decomposition methods for the heat equation in a space-time Lipschitz cylinder. The framework is based on a variational formulation, inspired by recent studies of space-time finite elements using Sobolev spaces with fractional time regularity. In this framework, the time-dependent Steklov--Poincar\'e operators are introduced and their essential properties are proven. We then derive the interface interpretations of the Dirichlet--Neumann, Neumann--Neumann and Robin--Robin methods and show that these methods are well defined. Finally, we prove convergence of the Robin--Robin method and introduce a modified method with stronger convergence properties.