# Convergence of Restricted additive Schwarz method with local impedance conditions for the Helmholtz equation

### Abstract

The Helmholtz equation is notoriously difficult to solve, especially for the case of high wavenumber $k$. The Restricted Additive Schwarz preconditioner (RAS) with local impedance problems is arguably the most successful one-level parallel method for Helmholtz problems. This preconditioner can be applied on very general geometries, does not require parameter-tuning, and can even be robust to increasing $k$. However, the standard preconditioning theory fails in predicting the performance of this RAS preconditioner: (1) the theory based on the condition number can not be applied to this case, since the discretized systems for Helmholtz problems are usually non-hermitian and indefinite; (2) the Elman’s theory based on the field of values (FOV) also fails, since the FOV of the RAS preconditioned matrix contains the original according to our computation. In this talk, we will present a new strategy based on the power contractivity’’ in the convergence analysis. We analyze the power of the iteration matrix directly and prove that it is a contraction if the norm of theimpedance-to-impedance maps’’ is small enough. We investigate the norm of these impedance maps using both PDE theory and FEM theory.

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