Summary: MULTIGRID PRECONDITIONING IN H(div)
ON NON-CONVEX POLYGONS*
DOUGLAS N. ARNOLD, RICHARD S. FALK, and RAGNAR WINTHER§
Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday.
Abstract. In an earlier paper we constructed and analyzed a multigrid preconditioner for the system
of linear algebraic equations arising from the finite element discretization of boundary value problems
associated to the differential operator I - grad div. In this paper we analyze the procedure without
assuming that the underlying domain is convex and show that, also in this case, the preconditioner is
spectrally equivalent to the inverse of the discrete operator.
Key words. preconditioner, finite element, multigrid, nonconvex domain
AMS(MOS) subject classifications (1991 revision). 65N55, 65N30
1. Introduction. In the earlier paper , we analyzed domain decomposition and
multigrid precondtioners for the efficient solution of the equations which arise from the
finite element discretization of boundary values problems for the operator I - grad div.
These results were then applied to construct efficient iterative methods for the solution
of the equations which arise from the finite element discretization of scalar second order
elliptic boundary value problems by mixed and least squares methods. In the case of the
domain decomposition algorithm, the convergence results were obtained first for the case
of a convex polygon, in which the solution of the scalar second order elliptic problem has