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Equivalence between lowest-order mixed finite elementand 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 elementmethod for second-order elliptic problems on simplicial meshes intwo and three space dimensions. This method produces saddle-pointproblems for scalar and flux unknowns. We show how to easily andlocally eliminate the flux unknowns, which implies the equivalencebetween this method and a particular multi-point finite volumescheme, without any approximate numerical integration. The matrixof the final linear system is sparse, positive definite for alarge class of problems, but in general nonsymmetric. We next showthat these ideas also apply to mixed and upwind-mixed finiteelement discretizations of nonlinear parabolicconvection–diffusion–reaction problems. Besides the theoreticalrelationship between the two methods, the results allow forimportant computational savings in the mixed finite elementmethod, which we finally illustrate on a set of numericalexperiments.

Keywords

Mixed finite element methodsaddle-point problemfinite volume methodsecond-order elliptic equationnonlinearparabolic 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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