A Coarse Space Formulation with good parallel properties for an Additive Schwarz Domain Decomposition Algorithm

Abstract

This paper studies a variant of the Additive Average Schwarz algorithm [1] where the coarse space consists of two subspaces. This approach results in a simpler and more parallel algorithm while we retain the essential convergence properties of the original method. Our theory is confirmed by numerical experiments showing that the new algorithm is often superior to the original variant. 1 Introduction In [1] two additive Schwarz algorithms were described and analyzed. The first, called the Additive Diagonal Scaling method, used a diagonal scaling derived from a coarse space V \Gamma1 (see below), combined with the classical coarse space V c 0 . This method was not robust when the jumps in the coefficients of the PDE could be arbitrarily distributed across subdomains. The second method, called the Additive Average Schwarz method, was proven robust, but uses a potentially expensive coarse space V avg 0 = Range(IA ) with I Au = ae u(x); x 2 @\Omega ih ¯ u i ; x 2\Omega ih ; ..