Optimized Schwarz Methods for Maxwell equations

Victorita Dolean (1, 2) , Martin Gander (3) , Luca Gerardo-Giorda (4)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 NACHOS - Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media
3 Section de mathématiques [Genève]
4 Department of mathematics/Dipartimento di Matematica [Univ. Trento]

Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. Over the last decade, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains, and are also convergent without overlap for elliptic problems. We show here why the classical Schwarz method applied to the hyperbolic problem converges without overlap for Maxwell's equations. The reason is that the method is equivalent to a simple optimized Schwarz method for an equivalent elliptic problem. Using this link, we show how to develop more efficient Schwarz methods than the classical ones for the Maxwell's equations. We illustrate our findings with numerical results.

Mots clés: Schwarz algorithms; optimized interface conditions; Maxwell equations

Domaines: Analyse numérique [math.NA]

Victorita Dolean, Martin Gander, Luca Gerardo-Giorda. Optimized Schwarz Methods for Maxwell equations. 2009. ⟨hal-00107263v5⟩