An explicit formulation of the multiplicative Schwarz preconditioner (Q2382752)

Suppose that \(A\) is a sparse \(n\times n\) matrix formed by \(k\) blocks arranged on the diagonal. Suppose that these blocks can possibly overlap at the corners. Let \(\mathbb C^n\) be divided into \(k\) ``subdomains'', which are defined to be the subspaces where each block acts. The purpose of the multiplicative Schwarz method is to iteratively solve linear systems of the form \(Ax=b\) in sequence on each of the subdomains. When the blocks are not overlapped, this method is equivalent to a block Gauss-Seidel iteration. The main result of this paper establishes an explicit formulation of the splitting in the multiplicative Schwarz method. Once the main result has been established, the authors show the advantages of using the method as a preconditioner of the Krylov method. Both theoretical and numerical arguments are presented to support this choice of preconditioner.