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Equivalence between lowest-order mixed finite element and multi-pointfinite volume methods on simplicial meshes

Published online by Cambridge University Press:  21 June 2006

Martin Vohralík
Martin Vohralík*
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic. vohralik@km1.fjfi.cvut.cz Laboratoire de Mathématiques, Analyse Numérique et EDP, Université de Paris-Sud, Bât. 425, 91405 Orsay, France. martin.vohralik@math.u-psud.fr
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Abstract

We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection–diffusion–reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments.

Keywords

Mixed finite element methodsaddle-point problemfinite volume methodsecond-order elliptic equationnonlinear parabolic convection–diffusion–reaction equation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 2 , March 2006 , pp. 367 - 391
Copyright
© EDP Sciences, SMAI, 2006

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