From face to element unknowns by local static condensation with application to nonconforming finite elements

We derive in this paper a new local static condensation strategy which allows to reduce significantly the number of unknowns in algebraic systems arising in discretization of partial differential equations. We apply it to the discretization of a model linear elliptic diffusion and a model nonlinear parabolic advection-diffusion-reaction problem by Crouzeix-Raviart nonconforming finite elements. Herein, the unknowns, originally associated with the mesh faces, can be reduced to new unknowns associated with the mesh elements. The resulting matrices are sparse, with possibly only four nonzero entries per row in two space dimensions, positive definite in dependence on the mesh geometry and the diffusion-dispersion tensor, but in general nonsymmetric. Our approach consists in introducing new element unknowns, the identification of suitable local vertex-based subproblems, and the inversion of the corresponding local matrices. We give sufficient conditions for the well-posedness of the local problems, as well as for the resulting global one. In addition, we provide a geometrical interpretation which suggests how to influence the form of the local and global matrices depending on the local mesh and data. We finally present an abstract generalization allowing for a further reduction of the number of unknowns, typically to one unknown per a set of mesh elements. We conclude by numerical experiments which show that the condition number of the resulting matrices is quite insensitive to the mesh anisotropies and the diffusion tensor inhomogeneities.