A Variational Interpretation of Restricted Additive Schwarz With Impedance Transmission Condition for the Helmholtz Problem

Abstract

Method (5)–(8) is an example of methods studied more generally in the Optimized Schwarz literature (e.g., [4, 10]), where Robin (or more sophisticated) transmission conditions are constructed with the aim of optimizing convergence rates. Although the transmission condition (6) above can be justified directly as a first order absorbing condition for the local Helmholtz problem (5) (without considering optimization), this method is still often called ‘OptimizedRestrictedAdditive Schwarz’ (or ‘ORAS’) and we shall continue this naming convention here. ORAS is arguably the most successful one-level parallel method for Helmholtz problems.