Implementation of Hybrid V-cycle Algebraic Multilevel 
            Methods for Mixed Finite Element Systems with Penalty

                               Chen-Y ao G. Lai

                         Department of Mathematics, 
                      National Chung-Cheng University, 
                                   No. 160,
             San-Hsing Village, Ming-Hsiung, Chia-Yi 621, Taiwan.
                        Email :  cylai@math.ccu.edu.tw

                                December 1994

                                   Abstract

The hybrid V-cycle algebraic multilevel methods proposed by Vassilevski [1]
are robust for solving scalar finite element equations of the second order, in
terms of the cost per iteration and the rate of convergence.  These methods
are based on the hierarchical decomposition of the finite element spaces
together with some inner iterations on certain levels of given (fixed)
multiplicity.  Our goal in this paper is the implementation of the hybrid
V-cycle hierarchical algebraic multilevel methods for the indefinite discrete
systems which arise when a mixed finite element approximation is used to solve
elliptic boundary value problems.

By introducing a penalty parameter, the perturbed indefinite system can be
reduced to a symmetric positive definite system containing the small penalty
parameter for the velocity unknown alone.  We use the hierarchical spatial
decomposition approach prop osed y Cai, Goldstein and Pasciak [2] for the
reduced system.  We demonstrate that the hybrid V-cycle multilevel iterative
scheme is uniformly convergent with respect to both the penalty parameter and
the number of discretization levels used.

                                  References

[1] P. S. Vassilevski.  Hybrid v-cycle algebraic multilevel preconditioners.
Math. Comp., 58:489-512, 1992.

[2] Z. Cai, C. Multilevel iteration for mixed finite element systems with
penalty.  SIAM J. Sci. Stat. Comput., 14:1072-1088, 1993.