Approximate Schur complement preconditioning of the lowest-order nodal discretizations

Certain classes of nodal methods and mixed-hybrid finite element methods lead to equivalent, robust, and accurate discretizations of second-order elliptic PDEs. However, widespread popularity of these discretizations has been hindered by the awkward linear systems which result. The present work overcomes this awkwardness and develops preconditioners which yield solution algorithms for these discretizations with an efficiency comparable to that of the multigrid method for standard discretizations. Our approach exploits the natural partitioning of the linear system obtained by the mixed-hybrid finite element method. By eliminating different subsets of unknowns, two Schur complements are obtained with known structure. Replacing key matrices in this structure by lumped 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 discretizations and hence treated adequately with only a single multigrid cycle. In this paper we restrict the discussion to the two-dimensional case; these techniques are readily extended to three dimensions.