Scientific computing and applications
Conjugate gradient method for dual-dual mixed formulations
Mathematics of Computation
On the numerical analysis of a nonlinear elliptic problem via mixed-FEM and Lagrange multipliers
Applied Numerical Mathematics
An upwind-mixed method on changing meshes for two-phase miscible flow in porous media
Applied Numerical Mathematics
A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support
Journal of Computational Physics
International Journal of Computer Mathematics
Applied Numerical Mathematics
A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Stability and error analysis of mixed finite-volume methods for advection dominated problems
Computers & Mathematics with Applications
Journal of Computational Physics
A Multipoint Flux Mixed Finite Element Method on Hexahedra
SIAM Journal on Numerical Analysis
Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods
Journal of Scientific Computing
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
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We present 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, the negative of its gradient, and its flux (the tensor coefficient 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 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 9-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 finite difference 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 flux 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.