Heterogeneous Optimized Schwarz Methods for Second Order Elliptic PDEs

Abstract

Due to their property of convergence in the absence of overlap, optimized Schwarz methods are the natural domain decomposition framework for heterogeneous problems, where the spatial decomposition is provided by the multiphysics of the phenomena. We study here heterogeneous problems which arise from the coupling of second order elliptic PDEs. Theoretical results and asymptotic formulas are proposed solving the corresponding min-max problems both for single and double sided optimizations, while numerical results confirm the effectiveness of our approach even when analytical conclusions are not available. Our analysis shows that optimized Schwarz methods do not suffer the heterogeneity, it is the opposite, they are faster the stronger the heterogeneity is. It is even possible to have h independent convergence choosing two independent Robin parameters. This property was proved for a Laplace equation with discontinuous coefficients, but only conjectured for more general couplings in [M. J. Gander and O. Dubois, Numer. Algorithms, 69 (2015), pp. 109--144]. Our study is completed by an application to a contaminant transport problem.

Mathematical Analysis and Simulation of Field Models in Accelerator Circuits

“…Intuitively, the optimised Schwarz waveform relaxation methods interpret the WR algorithm as a Schwarz method through time. Applying this to coupled systems arising from different models is sometimes called heterogeneous domain decomposition [152]. In Figure 5.3 a sketch of how the waveform relaxation algorithm of Figure 5.2 can be visualised as a domain decomposition method is given.…”

Patch-Smoother and Multigrid for the Dual Formulation for Linear Elasticity

“…The choice of using Robin, instead of Dirichlet or Neumann boundary conditions, in a domain decomposition approach for primal formulations, has been proved to be an efficient choice. See [10], [14], [12], [11], [27], [13]. In order to determine the relation between (37) and the case of discretized Robin conditions, let us consider, for the sake of simplicity, the strong formulation of the Poisson problem:…”

A numerical algorithm based on probing to find optimized transmission conditions

“…Optimized Schwarz Methods (OSMs) are very versatile: they can be used with or without overlap, converge faster compared to other domain decomposition methods [5], are among the fastest solvers for wave problems [9], and can be robust for heterogeneous problems [7]. This is due to their general transmission conditions, optimized for the problem at hand.…”