APPROXIMATE SCHUR COMPLEMENT PRECONDITIONING OF THE LOWEST-ORDER NODAL DISCRETIZATIONS

Certain classes of nodal methods and mixed-hybrid finite element metho ds lead to equivalent, robust, and accurate discretizations of second- order elliptic PDEs. However, widespread popularity of these discretiz ations has been hindered by the awkward linear systems which result. T he present work overcomes this awkwardness and develops preconditioner s which yield solution algorithms for these discretizations with an ef ficiency comparable to that of the multigrid method for standard discr etizations. Our approach exploits the natural partitioning of the line ar system obtained by the mixed-hybrid finite element method. By elimi nating different subsets of unknowns, two Schur complements are obtain ed with known structure. Replacing key matrices in this structure by l umped approximations, we define three optimal preconditioners. Central to the optimal performance of these preconditioners is their sparsity structure which is compatible with standard finite difference discret izations and hence treated adequately with only a single multigrid cyc le. In this paper we restrict the discussion to the two-dimensional ca se; these techniques are readily extended to three dimensions.