Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry

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Abstract
We present an expanded mixed finite element method for solving second-order elliptic partial differential equations on geometrically general domains. For the lowest-order Raviart--Thomas approximating spaces, we use quadrature rules to reduce the method to cell-centered finite differences, possibly enhanced with some face-centered pressures. This substantially reduces the computational complexity of the problem to a symmetric, positive definite system for essentially only as many unknowns as elements. Our new method handles general shape elements (triangles, quadrilaterals, and hexahedra) and full tensor coefficients, while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements. As in other mixed methods, we maintain the local approximation of the divergence (i.e., local mass conservation). In contrast, Galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation. Our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes. On nonsmooth meshes or with nonsmooth coefficients one can add Lagrange multiplier pressure unknowns on certain element edges or faces. This enhanced cell-centered procedure recovers full accuracy, with little additional cost if the coefficients or mesh geometry are piecewise smooth. Theoretical error estimates and numerical examples are given, illustrating the accuracy and efficiency of the methods.

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