PRECONDITIONING IN H(DIV) AND APPLICATIONS

We consider the solution of the system of linear algebraic equations w hich arises from the finite element discretization of boundary value p roblems associated to the differential operator I- grad div. The natur al setting for such problems is in the Hilbert space H(div) and the va riational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domai n decomposition and multigrid techniques. These preconditioners are sh own to be spectrally equivalent to the inverse of the operator. As a c onsequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of itera tions, with the number independent of the mesh discretization. We desc ribe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary v alue problems.