Finite Volume Box Schemes and Mixed Methods

Jean-Pierre Croisille

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 5, page 1087-1106
  • ISSN: 0764-583X

Abstract

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We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form div ϕ ( u , u ) = f . The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u and the div-conforming space of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.

How to cite

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Croisille, Jean-Pierre. "Finite Volume Box Schemes and Mixed Methods." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 1087-1106. <http://eudml.org/doc/197549>.

@article{Croisille2010,
abstract = { We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathop\{\rm div\}\nolimits\varphi(u,\nabla u)=f$. The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u and the div-conforming space of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method. },
author = {Croisille, Jean-Pierre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Box method; box scheme; mixed finite element method; Petrov-Galerkin method; finite volume method.; Poisson equation; mixed Petrov-Galerkin finite volume schemes; error estimate},
language = {eng},
month = {3},
number = {5},
pages = {1087-1106},
publisher = {EDP Sciences},
title = {Finite Volume Box Schemes and Mixed Methods},
url = {http://eudml.org/doc/197549},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Croisille, Jean-Pierre
TI - Finite Volume Box Schemes and Mixed Methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 5
SP - 1087
EP - 1106
AB - We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathop{\rm div}\nolimits\varphi(u,\nabla u)=f$. The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u and the div-conforming space of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.
LA - eng
KW - Box method; box scheme; mixed finite element method; Petrov-Galerkin method; finite volume method.; Poisson equation; mixed Petrov-Galerkin finite volume schemes; error estimate
UR - http://eudml.org/doc/197549
ER -

References

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