An introduction to Multitrace Formulations and Associated Domain Decomposition Solvers

TL;DR

This paper introduces multitrace formulations (MTFs) for domain decomposition, providing new insights, optimal parameters, and connections to Schwarz methods, supported by theoretical analysis and numerical experiments.

Contribution

It offers a simplified, accessible introduction to MTFs, reveals their relation to Schwarz methods, and derives geometry-independent convergence results and optimal relaxation parameters.

Findings

Optimal relaxation parameters for block Jacobi iteration are determined.

Multitrace formulations relate to optimal Schwarz methods using Dirichlet to Neumann maps.

Convergence results are independent of geometry, dimension, and elliptic operator form.

Abstract

Multitrace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. We introduce in this paper MTFs on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multitrace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderon projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem{\color{black}, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.