Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, p. 367-391

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.

DOI : https://doi.org/10.1051/m2an:2006013
Classification:  76M10,  76M12,  76S05
Keywords: mixed finite element method, saddle-point problem, finite volume method, second-order elliptic equation, nonlinear parabolic convection-diffusion-reaction equation
@article{M2AN_2006__40_2_367_0,
author = {Vohral\'\i k, Martin},
title = {Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
pages = {367-391},
doi = {10.1051/m2an:2006013},
zbl = {1116.65121},
mrnumber = {2241828},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_2_367_0}
}

Vohralík, Martin. Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, pp. 367-391. doi : 10.1051/m2an:2006013. http://www.numdam.org/item/M2AN_2006__40_2_367_0/

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