Robust Additive Schwarz Preconditioners for Abstract Symmetric Positive Definite Operators

Dr. Jörg Willems

Dec. 7, 2010, 3:30 p.m. P 004

A framework for constructing robust additive Schwarz preconditioners for general symmetric positive definite (SPD) problems is presented. The term robustrefers to the property of the condition numbers of the preconditioned systems being independent of mesh parameters and problem parameters. Important instances of such problem parameters are e.g. (highly varying) conductivities and permeabilities. The core of this method is the construction of the coarse space based on the solution of local generalized eigenvalue problems. The framework only requires assumptions which are naturally satisfied by SPD operators corresponding to partial differential equations and is thus applicable to a wide range of problems. To address a broader audience we provide some general background information on domain decomposition methods and discuss the application of our framework to the scalar elliptic equation with jumping coefficients.