Preconditioners for Schwarz relaxation methods applied to differential algebraic equations

Abstract

In this paper, the authors investigate the ability of Schwarz relaxation (SR) methods to deal with large systems of differential algebraic equations (DAEs) and assess their respective efficiency. Since the number of iterations required to achieve convergence of the classical SR method is strongly related to the number of subdomains and the time step size, two new preconditioning techniques are here developed. A preconditioner based on a correction using the algebraic equations is first introduced and leads to a number of iterations independent on the number of subdomains. A second preconditioner based on a correction using the Schur complement matrix makes the convergence independent on both the number of subdomains and the integration step size. Application on European electricity network is presented to outline the performance, efficiency, and robustness of the proposed preconditioning techniques for the solution of DAEs.