Preconditioning discretizations of systems of partial differential equations

TLDR

This survey reviews preconditioner construction for symmetric linear systems arising from PDE saddle‑point problems, drawing on talks presented at the 2008 London Mathematical Society Durham Symposium and prior literature. The paper proposes an abstract Hilbert‑space framework for constructing block‑diagonal preconditioners for symmetric linear systems, illustrating the approach with unified examples. The framework is applied to parameter‑dependent systems from incompressible flow, linear elasticity, and optimal control theory. © 2010 John Wiley & Sons, Ltd.

Abstract

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter-dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach. Copyright © 2010 John Wiley & Sons, Ltd.