On a stabilized colocated Finite Volume scheme for the Stokes problem

Robert Eymard; Raphaèle Herbin; Jean Claude Latché

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 3, page 501-527
  • ISSN: 0764-583X

Abstract

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We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem. It has been developed following two main ideas. On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes. On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique. We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order. With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions. Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

How to cite

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Eymard, Robert, Herbin, Raphaèle, and Latché, Jean Claude. "On a stabilized colocated Finite Volume scheme for the Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis 40.3 (2006): 501-527. <http://eudml.org/doc/249711>.

@article{Eymard2006,
abstract = { We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem. It has been developed following two main ideas. On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes. On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique. We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order. With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions. Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm. },
author = {Eymard, Robert, Herbin, Raphaèle, Latché, Jean Claude},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite volumes; colocated discretizations; Stokes problem; incompressible flows; analysis.; finite volumes; collocated discretizations; analysis},
language = {eng},
month = {7},
number = {3},
pages = {501-527},
publisher = {EDP Sciences},
title = {On a stabilized colocated Finite Volume scheme for the Stokes problem},
url = {http://eudml.org/doc/249711},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Eymard, Robert
AU - Herbin, Raphaèle
AU - Latché, Jean Claude
TI - On a stabilized colocated Finite Volume scheme for the Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 3
SP - 501
EP - 527
AB - We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem. It has been developed following two main ideas. On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes. On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique. We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order. With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions. Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.
LA - eng
KW - Finite volumes; colocated discretizations; Stokes problem; incompressible flows; analysis.; finite volumes; collocated discretizations; analysis
UR - http://eudml.org/doc/249711
ER -

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