An introduction to Multitrace Formulations and Associated Domain Decomposition Solvers
For researchers in domain decomposition and computational science, this work provides a simplified understanding of MTFs and establishes their connection to optimal Schwarz methods, though it is incremental as it builds on existing concepts.
This paper introduces multitrace formulations (MTFs) on a simple model problem to make them accessible to domain decomposition researchers, revealing a natural block Jacobi iteration with optimal relaxation parameters and showing that iterative MTF solvers relate to the optimal Schwarz method. The convergence results and optimal relaxation parameter are independent of geometry, dimension, and elliptic operator, as demonstrated by numerical experiments.
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.