Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes

Abdallah Bradji; Jürgen Fuhrmann

Applications of Mathematics (2013)

  • Volume: 58, Issue: 1, page 1-38
  • ISSN: 0862-7940

Abstract

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A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. We derive error estimates in discrete norms 𝕃 ∞ ( 0 , T ; H 0 1 ( Ω ) ) and 𝒲 1 , ∞ ( 0 , T ; L 2 ( Ω ) ) , and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form involved in the finite volume scheme satisfies some ellipticity condition.

How to cite

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Bradji, Abdallah, and Fuhrmann, Jürgen. "Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes." Applications of Mathematics 58.1 (2013): 1-38. <http://eudml.org/doc/251385>.

@article{Bradji2013,
abstract = {A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. We derive error estimates in discrete norms $\mathbb \{L\}^\{\infty \}(0,T;H^1_0(\Omega ))$ and $\{\mathcal \{W\}\}^\{1,\infty \}(0,T;L^2(\Omega ))$, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form involved in the finite volume scheme satisfies some ellipticity condition.},
author = {Bradji, Abdallah, Fuhrmann, Jürgen},
journal = {Applications of Mathematics},
keywords = {non-conforming grid; nonstationary heat equation; several space dimension; SUSHI scheme; implicit scheme; discrete gradient; finite volume method; heat equation; error estimate; discrete gradient; nonconforming grids; star-shaped elements; SUSHI scheme},
language = {eng},
number = {1},
pages = {1-38},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes},
url = {http://eudml.org/doc/251385},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Bradji, Abdallah
AU - Fuhrmann, Jürgen
TI - Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 1
EP - 38
AB - A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. We derive error estimates in discrete norms $\mathbb {L}^{\infty }(0,T;H^1_0(\Omega ))$ and ${\mathcal {W}}^{1,\infty }(0,T;L^2(\Omega ))$, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form involved in the finite volume scheme satisfies some ellipticity condition.
LA - eng
KW - non-conforming grid; nonstationary heat equation; several space dimension; SUSHI scheme; implicit scheme; discrete gradient; finite volume method; heat equation; error estimate; discrete gradient; nonconforming grids; star-shaped elements; SUSHI scheme
UR - http://eudml.org/doc/251385
ER -

References

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