Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation

Francis Collino (1) , Patrick Joly (1) , Matthieu Lecouvez (2)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 CESTA - Centre d'études des systèmes et des technologies avancées

In this paper, we develop in a general framework a non overlapping Domain Decomposition Method that is proven to be well-posed and converges exponentially fast, provided that specific transmission operators are used. These operators are necessarily non local and we provide a class of such operators in the form of integral operators. To reduce the numerical cost of these integral operators, we show that a truncation process can be applied that preserves all the properties leading to an exponentially fast convergent method. A modal analysis is performed on a separable geometry to illustrate the theoretical properties of the method and we exhibit an optimization process to further reduce the convergence rate of the algorithm.

Keywords: Domain decomposition methods; exponentially fast convergent methods; integral operators; norms of fractional order Sobolev spaces; pseudo-differential operators.

Domains: Mathematics [math]

Francis Collino, Patrick Joly, Matthieu Lecouvez. Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation. ESAIM: Mathematical Modelling and Numerical Analysis, 2020, 54 (3), pp.775-810. ⟨10.1051/m2an/2019050⟩. ⟨cea-03052206⟩