Time-dependent Steklov--Poincar\'e operators and space-time Robin--Robin decomposition for the heat equation

TL;DR

This paper develops a new variational framework for analyzing space-time domain decomposition methods for the heat equation, introducing time-dependent Steklov--Poincaré operators and proving convergence of Robin--Robin methods.

Contribution

It introduces a novel variational framework for parabolic PDEs, defining time-dependent Steklov--Poincaré operators and analyzing Robin--Robin domain decomposition methods.

Findings

Defined time-dependent Steklov--Poincaré operators and proved their properties.

Derived interface interpretations for classical domain decomposition methods.

Proved convergence of the Robin--Robin method and proposed a more robust variant.

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.