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Enhanced cell-centered finite differences for elliptic equations on general geometry. (English) Zbl 0947.65114

An “expanded” mixed finite element method for second-order elliptic problems is presented and studied. This means that the pressure equation \[ -\nabla\cdot(k\nabla p)= f \] is presented in a mixed form expanded with one more unknown vector variable \(\widetilde{\mathbf u}\): \[ G\widetilde{\mathbf u}= -\nabla p,\quad {\mathbf u}= KG\widetilde{\mathbf u},\quad \nabla\cdot{\mathbf u}= f. \] A key point in such formulation is a choice of the matrix \(G\), together with an appropriate quadrature formula, so that the mixed finite element method for the expanded system produces cell-centered finite difference approximation for the pressure equation for both triangles and quadrilaterals. The obtained schemes are locally conservative and retain the optimal order of convergence and are also superconvergent under additional assumptions on the mesh.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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