A unified analysis of elliptic problems with various boundary conditions and their approximation

Jérôme Droniou; Robert Eymard; Thierry Gallouët; Raphaèle Herbin

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 2, page 339-368
  • ISSN: 0011-4642

Abstract

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We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming (that is, the approximation functions can belong to the energy space relative to the problem) or not, and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to some models such as flows in fractured medium, elasticity equations and diffusion equations on manifolds.

How to cite

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Droniou, Jérôme, et al. "A unified analysis of elliptic problems with various boundary conditions and their approximation." Czechoslovak Mathematical Journal 70.2 (2020): 339-368. <http://eudml.org/doc/297097>.

@article{Droniou2020,
abstract = {We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming (that is, the approximation functions can belong to the energy space relative to the problem) or not, and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to some models such as flows in fractured medium, elasticity equations and diffusion equations on manifolds.},
author = {Droniou, Jérôme, Eymard, Robert, Gallouët, Thierry, Herbin, Raphaèle},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic problem; various boundary conditions; gradient discretisation method; Leray-Lions problem},
language = {eng},
number = {2},
pages = {339-368},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A unified analysis of elliptic problems with various boundary conditions and their approximation},
url = {http://eudml.org/doc/297097},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Droniou, Jérôme
AU - Eymard, Robert
AU - Gallouët, Thierry
AU - Herbin, Raphaèle
TI - A unified analysis of elliptic problems with various boundary conditions and their approximation
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 2
SP - 339
EP - 368
AB - We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming (that is, the approximation functions can belong to the energy space relative to the problem) or not, and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to some models such as flows in fractured medium, elasticity equations and diffusion equations on manifolds.
LA - eng
KW - elliptic problem; various boundary conditions; gradient discretisation method; Leray-Lions problem
UR - http://eudml.org/doc/297097
ER -

References

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