On the convergence of multigrid iteration scheme for mixed finite element equation

Abstract

We present a multigrid iteration algorithm for mixed finite element equations and a proof of its convergence where the rate of convergence is independent of the mesh size. Whereas techniques such as the “multigrid preconditioning” and “non-nested multigrid schemes” have been recently proposed to solve the indefinite linear system arising from the mixed finite element approximation, our analysis here clearly indicates that the multigrid idea for solving linear systems could be applied directly to mixed finite element subspace. We hope that this fact encourages one to consider multigrid solution for other kinds of non-conforming finite element equations for the elliptic PDEs. To establish our convergence theorem, we derive and apply an energy inequality that is weaker than some norm estimates used by others; in fact, such norm estimates are not applicable to establish our convergence result.