Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Robert Eymard; Raphaèle Herbin; Jean-Claude Latché; Bruno Piar

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 5, page 889-927
  • ISSN: 0764-583X

Abstract

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We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory 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, et al. "Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows." ESAIM: Mathematical Modelling and Numerical Analysis 43.5 (2009): 889-927. <http://eudml.org/doc/250597>.

@article{Eymard2009,
abstract = { We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm. },
author = {Eymard, Robert, Herbin, Raphaèle, Latché, Jean-Claude, Piar, Bruno},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite volumes; collocated discretizations; Stokes problem; Navier-Stokes equations; incompressible flows; analysis; finite volumes},
language = {eng},
month = {8},
number = {5},
pages = {889-927},
publisher = {EDP Sciences},
title = {Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows},
url = {http://eudml.org/doc/250597},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Eymard, Robert
AU - Herbin, Raphaèle
AU - Latché, Jean-Claude
AU - Piar, Bruno
TI - Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/8//
PB - EDP Sciences
VL - 43
IS - 5
SP - 889
EP - 927
AB - We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.
LA - eng
KW - Finite volumes; collocated discretizations; Stokes problem; Navier-Stokes equations; incompressible flows; analysis; finite volumes
UR - http://eudml.org/doc/250597
ER -

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