Browsing by Author "Yotov, Ivan"

Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Finite Differences
    (1994-01) Arbogast, Todd; Wheeler, Mary F.; Yotov, Ivan
    We develop the theory of an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, its gradient, and its flux (the tensor coefficient times the gradient). The expected optimal order approximations are obtained in the L² and H^{-s}-norms, and superconvergence is obtained between the L²-projection of the scalar variable and its approximation. The scheme is suitable for the case in which the coefficient is a tensor that may have zeros, since it does not need to be inverted.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 finite difference method, requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. We show that the optimal rates of convergence are retained; moreover, superconvergence is obtained for the scalar unknown as well as for its gradient and flux at certain discrete points. Computational results illustrate these theoretical results.