Partially Updated Restricted Additive Schwarz Preconditioner

Abstract

The solution of differential equations with implicit methods requires the solution of a nonlinear problem at each time step. We consider Newton-Krylov ([8], Chap. 3) methods to solve these nonlinear problems: the linearized system of each Newton iteration of each time step is solved by a Krylov method. Generally speaking, the most time-consuming part of the numerical simulation is the solution of the sequence of linear systems by the Krylov method. Then, providing a good preconditioner is a critical point: a balance must be found between the ability of the preconditioner to reduce the number of Krylov iterations, and its computational cost. The method that combines a Newton-Krylov method with a Schwarz domain decomposition preconditioner is called Newton-Krylov-Schwarz (NKS) [5]. In this paper, we deal with the Restricted Additive Schwarz (RAS) preconditioner [4]. We propose to freeze this preconditioner for a few time steps, and to partially update it. Here, the partial update of the preconditioner consists in recomputing some parts of the preconditioner associated to certain subdomains, keeping the other ones frozen. These partial updates improve the efficiency and the longevity of the frozen preconditioner. Furthermore, they can be computed asynchronously in order to improve the parallelism.