Mixed Finite Elements for Elliptic Problems with Tensor Coeecients as Cell-centered Finite Diierences Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-centered Finite Differences

We present an expanded mixed nite element approximation of second order elliptic problems containing a tensor coeecient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its ux (the tensor coeecient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart-Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered nite diierence method, requiring the solution of a sparse, positive semideenite linear system for the scalar unknown. For a general tensor coeecient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the L 2 and H ?s-norms (and superconvergence is obtained between the L 2-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the nite diierence method. If h denotes the maximal mesh spacing, then the optimal rate is O(h). The superconvergence rate O(h 2) is obtained for the scalar unknown and rate O(h 3=2) for its gradient and ux in certain discrete norms; moreover, the full O(h 2) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.