Geodesic
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

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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 : 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 
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     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},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {2},
     year = {2006},
     doi = {10.1051/m2an:2006013},
     mrnumber = {2241828},
     zbl = {1116.65121},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006013/}
}
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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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