Heterogeneous Optimized Schwarz Methods for Coupling Helmholtz and Laplace Equations
Optimized Schwarz methods have increasingly drawn attention over the last decades because of their improvements in terms of robustness and computational cost compared to the classical Schwarz method. Optimized Schwarz methods are also a natural framework to study heterogeneous phenomena, where the spatial decomposition is provided by the multi-physics of the problem, because of their good convergence properties in the absence of overlap. We propose here zeroth order optimized transmission conditions for the coupling between the Helmholtz equation and the Laplace equation, giving asymptotically optimized choices for the parameters, and illustrating our analytical results with numerical experiments.
The authors are grateful to L. Halpern for very useful remarks concerning the well posedness analysis.
- 1.P. Ciarlet, Linear and Nonlinear Functional Analysis with Applications: Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 2013)Google Scholar
- 2.B. Després, Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régimes harmoniques. Ph.D. thesis, Université Dauphine-Parix IX, 1991Google Scholar
- 4.M. El Bouajaji, V. Dolean, M.J. Gander, S. Lanteri, Comparison of a one and two parameter family of transmission conditions for Maxwells equations with damping, in Domain Decomposition Methods in Science and Engineering XX (Springer, Berlin, 2013), pp. 271–278Google Scholar
- 9.P. Grisvard, Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 1985)Google Scholar