Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities

An abstract setting for robustly preconditioning symmetric positive definite (SPD) operators is presented. The term “robust” refers 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 in particular (highly varying) coefficients. The method belongs to the class of additive Schwarz preconditioners. The core of the method is a robust stable decomposition of functions into several local and one coarse component. The coarse component is contained in a coarse space whose construction is based on the solution of local generalized eigenvalue problems. The paper gives an overview of the results obtained in a recent paper by the authors. It, furthermore, focuses on the importance of weighted Poincaré inequalities (WPIs) for the analysis of stable decompositions. WPIs have recently received attention in the setting of the scalar elliptic case. In our abstract framework we extend the notion of WPIs to general SPD operators. To demonstrate the applicability of the abstract preconditioner the scalar elliptic equation and the stream function formulation of Brinkman’s equations in two spatial dimensions are considered. Several numerical examples are presented.