A Primal-Based Penalty Preconditioner for Elliptic Saddle Point Systems

Abstract

A primal-based penalty preconditioner is presented for a linear set of equations arising from elliptic saddle point problems. We show that the eigenvalues of the preconditioned matrix are positive real and demonstrate that a variant of the preconditioner can be combined with the conjugate gradient algorithm. Our approach is motivated by two basic observations. First, the solution of a problem with constraints is often similar to the solution of a problem where the constraints are penalized. Second, certain methods of solution not available for a constrained problem are possible for its penalized counterpart so motivating a primal-based Schur complement approach. Numerical examples for elliptic two- and three-dimensional problems are presented that confirm theoretical results and demonstrate the effectiveness of the preconditioner.