MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS*

Abstract. In an earlier paper we constructed and analyzed a multigrid preconditioner for the systemof linear algebraic equations arising from the finite element discretization of boundary value problemsassociated to the differential operator I− graddiv. In this paper we analyze the procedure withoutassuming that the underlying domain is convex and show that, also in this case, the preconditioner isspectrally equivalent to the inverse of the discrete operator.Key words. preconditioner, finite element, multigrid, nonconvex domainAMS(MOS) subject classifications (1991 revision). 65N55, 65N30 1. Introduction. In the earlier paper [1], we analyzed domain decomposition andmultigrid precondtioners for the efficient solution of the equations which arise from thefinite element discretization of boundary values problems for the operator I− graddiv.These results were then applied to construct efficient iterative methods for the solutionof the equations which arise from the finite element discretization of scalar second orderelliptic boundary value problems by mixed and least squares methods. In the case of thedomain decomposition algorithm, the convergence results were obtained first for the caseof a convex polygon, in which the solution of the scalar second order elliptic problem hasH