Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type

Clément Cancès; Jérôme Droniou; Cindy Guichard; Gianmarco Manzini; Manuela Bastidas Olivares; Iuliu Sorin Pop

doi:10.1007/978-3-030-69363-3_2

Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type

Clément Cancès
, Jérôme Droniou
, Cindy Guichard
, Gianmarco Manzini
, Manuela Bastidas Olivares
, Iuliu Sorin Pop

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter (Book) › Research › peer-review

3 Citations (Scopus)

Abstract

The gradient discretisation method (GDM) is a generic framework for the spatial discretisation of partial differential equations. The goal of this contribution is to establish an error estimate for a class of degenerate parabolic problems, obtained under very mild regularity assumptions on the exact solution. Our study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results.

Original language	English
Title of host publication	Polyhedral Methods in Geosciences
Editors	Daniele Antonio Di Pietro , Luca Formaggia, Roland Masson
Place of Publication	Cham Switzerland
Publisher	Springer
Chapter	2
Pages	37-72
Number of pages	36
Volume	27
ISBN (Electronic)	9783030693633
ISBN (Print)	9783030693626
DOIs	https://doi.org/10.1007/978-3-030-69363-3_2
Publication status	Published - 2021

Publication series

Name	SEMA SIMAI Springer Series
Volume	27
ISSN (Print)	2199-3041
ISSN (Electronic)	2199-305X

Keywords

Discontinuous Galerkin method
Error estimates
Fast diffusion
Gradient discretisation method
Hybrid mimetic mixed method
Numerical tests
Polytopal methods
Porous medium equation
Slow diffusion
Vertex approximate gradient method
Virtual element method

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10.1007/978-3-030-69363-3_2Licence: CC BY

Cite this

Cancès, C., Droniou, J., Guichard, C., Manzini, G., Olivares, M. B., & Pop, I. S. (2021). Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type. In D. Antonio Di Pietro , L. Formaggia, & R. Masson (Eds.), Polyhedral Methods in Geosciences (Vol. 27, pp. 37-72). (SEMA SIMAI Springer Series; Vol. 27). Springer. https://doi.org/10.1007/978-3-030-69363-3_2

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abstract = "The gradient discretisation method (GDM) is a generic framework for the spatial discretisation of partial differential equations. The goal of this contribution is to establish an error estimate for a class of degenerate parabolic problems, obtained under very mild regularity assumptions on the exact solution. Our study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results.",

keywords = "Discontinuous Galerkin method, Error estimates, Fast diffusion, Gradient discretisation method, Hybrid mimetic mixed method, Numerical tests, Polytopal methods, Porous medium equation, Slow diffusion, Vertex approximate gradient method, Virtual element method",

author = "Cl{\'e}ment Canc{\`e}s and J{\'e}r{\^o}me Droniou and Cindy Guichard and Gianmarco Manzini and Olivares, \{Manuela Bastidas\} and Pop, \{Iuliu Sorin\}",

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Cancès, C, Droniou, J, Guichard, C, Manzini, G, Olivares, MB & Pop, IS 2021, Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type. in D Antonio Di Pietro , L Formaggia & R Masson (eds), Polyhedral Methods in Geosciences. vol. 27, SEMA SIMAI Springer Series, vol. 27, Springer, Cham Switzerland, pp. 37-72. https://doi.org/10.1007/978-3-030-69363-3_2

Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type. / Cancès, Clément; Droniou, Jérôme; Guichard, Cindy et al.
Polyhedral Methods in Geosciences. ed. / Daniele Antonio Di Pietro ; Luca Formaggia; Roland Masson. Vol. 27 Cham Switzerland: Springer, 2021. p. 37-72 (SEMA SIMAI Springer Series; Vol. 27).

Research output: Chapter in Book/Report/Conference proceeding › Chapter (Book) › Research › peer-review

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T1 - Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type

AU - Cancès, Clément

AU - Droniou, Jérôme

AU - Guichard, Cindy

AU - Manzini, Gianmarco

AU - Olivares, Manuela Bastidas

AU - Pop, Iuliu Sorin

PY - 2021

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N2 - The gradient discretisation method (GDM) is a generic framework for the spatial discretisation of partial differential equations. The goal of this contribution is to establish an error estimate for a class of degenerate parabolic problems, obtained under very mild regularity assumptions on the exact solution. Our study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results.

AB - The gradient discretisation method (GDM) is a generic framework for the spatial discretisation of partial differential equations. The goal of this contribution is to establish an error estimate for a class of degenerate parabolic problems, obtained under very mild regularity assumptions on the exact solution. Our study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results.

KW - Discontinuous Galerkin method

KW - Error estimates

KW - Fast diffusion

KW - Gradient discretisation method

KW - Hybrid mimetic mixed method

KW - Numerical tests

KW - Polytopal methods

KW - Porous medium equation

KW - Slow diffusion

KW - Vertex approximate gradient method

KW - Virtual element method

UR - https://www.scopus.com/pages/publications/85108441419

U2 - 10.1007/978-3-030-69363-3_2

DO - 10.1007/978-3-030-69363-3_2

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BT - Polyhedral Methods in Geosciences

A2 - Antonio Di Pietro , Daniele

A2 - Formaggia, Luca

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Cancès C, Droniou J, Guichard C, Manzini G, Olivares MB, Pop IS. Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type. In Antonio Di Pietro D, Formaggia L, Masson R, editors, Polyhedral Methods in Geosciences. Vol. 27. Cham Switzerland: Springer. 2021. p. 37-72. (SEMA SIMAI Springer Series). doi: 10.1007/978-3-030-69363-3_2

Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type

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