Multilevel Iteration for Mixed Finite Element Systems with Penalty

In this paper, the authors consider the solution of the discrete systems that arises when a mixed finite element approach is used to approximate the solution of second-order elliptic boundary value problems. By the introduction of a penalty parameter, these equations can be approximated by the solution of a symmetric and positive definite penalty system on the velocity subspace. Iterative procedures are developed and analyzed for this penalty system based on the hierarchical basis approach as well as on the standard multigrid approach. Finally, numerical experiments are presented that illustrate the convergence behavior suggested by the theory.