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An introduction to multitrace formulations and associated domain decomposition solvers

Claeys, X. and Dolean, V. and Gander, M. J. (2019) An introduction to multitrace formulations and associated domain decomposition solvers. Applied Numerical Mathematics, 135. pp. 69-86. ISSN 0168-9274

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Abstract

Multi-trace 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. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations 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 multi-trace 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 Calderón 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, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.