Abstract
Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming finite element, mixed finite element and finite volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards’, Stefan’s and Leray–Lions’ models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli–Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin–Simon theorem. The model’s degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.
| Original language | English |
|---|---|
| Pages (from-to) | 721-766 |
| Number of pages | 46 |
| Journal | Numerische Mathematik |
| Volume | 132 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 1 Apr 2016 |
Keywords
- 35K65
- 46N40
- 65M12
Cite this
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Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations. / Droniou, Jerome; Eymard, Robert.
In: Numerische Mathematik, Vol. 132, No. 4, 01.04.2016, p. 721-766.Research output: Contribution to journal › Article › Research › peer-review
TY - JOUR
T1 - Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations
AU - Droniou, Jerome
AU - Eymard, Robert
PY - 2016/4/1
Y1 - 2016/4/1
N2 - Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming finite element, mixed finite element and finite volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards’, Stefan’s and Leray–Lions’ models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli–Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin–Simon theorem. The model’s degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.
AB - Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming finite element, mixed finite element and finite volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards’, Stefan’s and Leray–Lions’ models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli–Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin–Simon theorem. The model’s degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.
KW - 35K65
KW - 46N40
KW - 65M12
UR - http://www.scopus.com/inward/record.url?scp=84961215335&partnerID=8YFLogxK
U2 - 10.1007/s00211-015-0733-6
DO - 10.1007/s00211-015-0733-6
M3 - Article
VL - 132
SP - 721
EP - 766
JO - Numerische Mathematik
JF - Numerische Mathematik
SN - 0029-599X
IS - 4
ER -