Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems

Abstract

We discuss certain preconditioning techniques for solving indefinite linear systems of equations arising from mixed finite element discretizations of elliptic equations of second order. The techniques are based on various approximations of the mass matrix, say, by simply lumping it to be diagonal or by constructing a diagonal matrix assembled of properly scaled lumped element mass matrices. We outline two possible alternatives for preconditioning. One can precondition the original (indefinite) system by some indefinite matrix and hence use either a stationary iterative method or a generalized conjugate gradient type method. Alternatively as in the particular case of rectangular Raviart-Thomas elements, which we consider, one can perform iterations in a subspace, eliminating the velocity unknowns and then considering the corresponding reduced system which is elliptic. So we can use the ordinary preconditioned conjugate gradient method and any known preconditioner (of optimal order, for example, like the multigrid method) for the corresponding finite element discretization of the elliptic problem. Numerical experiments for some of the proposed iterative methods are presented.Keywordsindefinite systempreconditioningiterations in subspaceconjugate gradientsmixed finite elementssecond order elliptic problemsSubject ClassificationsAMS(MOS) 65F1065N2065N30