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 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 },
pages = {367--391},
year = {2006},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
doi = {10.1051/m2an:2006013},
mrnumber = {2241828},
zbl = {1116.65121},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006013/}
}
TY - JOUR AU - Vohralík, Martin TI - Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2006 SP - 367 EP - 391 VL - 40 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006013/ DO - 10.1051/m2an:2006013 LA - en ID - M2AN_2006__40_2_367_0 ER -
%0 Journal Article %A Vohralík, Martin %T Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2006 %P 367-391 %V 40 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006013/ %R 10.1051/m2an:2006013 %G en %F 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 , Tome 40 (2006) no. 2, pp. 367-391. doi: 10.1051/m2an:2006013
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