Preconditioning for regular elliptic systems of differential equations

John C. Strikwerda
Department of Computer Sciences
University of Wisconsin-Madison

Abstract: In solving linear systems of equations arising from discretizations of elliptic systems of equations, it is common to precondition the system so that the resulting system is easier to solve by an iterative method. What this means is that the linear system Ax=b is replaced by the system PA x = Pb, where P, the preconditioning matrix, is chosen to mimic the inverse of A in some way. For example, the eigenvalues of PA may be clustered much more closely than the eigenvalues of A itself. If A results for the discretization of an elliptic differential operator E, then one way to choose P is as an approximation to the inverse of a differential operator G so that G-1 E is a bounded operator. For example, if E is a second-order elliptic operator with variable coefficients, then G might be the Laplacian. This approach then considers preconditioning for elliptic operators, rather than for matrices, allowing for a rigorous analysis of this type of preconditioners. In this talk I will present a general theory for preconditioning elliptic systems, especially for the Stokes equations and the equations of linear elasticity.