Convergence of iterative solvers for the Helmholtz equation

The Restricted Additive Schwarz method with impedance transmission conditions (often called the ORAS method) is a domain decomposition method which can be used as an iterative solver or as a preconditioner for the solution of discretized Helmholtz boundary-value problems. It is a very simple parallel one-level method, applicable in very general geometries and it has been successfully combined with coarse spaces to obtain two- or multi-level methods. Its multiplicative variants are related to 'sweeping methods' which have enjoyed considerable recent practical interest. To date there is relatively little convergence analysis for this method. In the talk we present a novel analysis of the ORAS method.

The main components of the talk are:
(i) ORAS is a non-conforming finite element approximation of a classical parallel Schwarz method formulated at the PDE level;
(ii) The parallel Schwarz method is well-posed in suitable function spaces of Helmholtz-harmonic functions in general geometries;
(iii) The parallel Schwarz method is proved to be power contractive for domain decompositions of 'strip-type' and is observed to be power contractive for general domain decompositions in 2D experiments;
(iv) Working in suitable Helmholtz-harmonic finite element spaces and for fine enough meshes, the ORAS method is proved to enjoy the same convergence estimates as the parallel Schwarz method; in particular its power contractive property is independent of the polynomial order of the finite element spaces used.

The proof of (iv) uses a new finite element error estimate, proving higher order convergence for certain Helmholtz problems at interior interfaces.

The work on ORAS is recent joint work with Shihua Gong and Euan Spence (Bath) while the results on the parallel Schwarz method were obtained with Shihua Gong, Martin Gander (Geneva), David Lafontaine (Bath) and Euan Spence.