A mixed finite element method for nonlinear diffusion equations

Handle URI:
http://hdl.handle.net/10754/597308
Title:
A mixed finite element method for nonlinear diffusion equations
Authors:
Burger, Martin; Carrillo, José; Wolfram, Marie-Therese
Abstract:
We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.
Citation:
Burger M, Carrillo J, Wolfram M-T (2010) A mixed finite element method for nonlinear diffusion equations. KRM 3: 59–83. Available: http://dx.doi.org/10.3934/krm.2010.3.59.
Publisher:
American Institute of Mathematical Sciences (AIMS)
Journal:
Kinetic and Related Models
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Jan-2010
DOI:
10.3934/krm.2010.3.59
Type:
Article
ISSN:
1937-5093
Sponsors:
JAC was partially supported by MTM2008-06349-C03-03 project DGI-MCI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. MTW acknowledges support by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). We acknowledge the Institute for Pure and Applied Mathematics (University of California, Los Angeles), the International Center for Mathematical Sciences (Edinburgh, UK), the Centro de Ciencias de Benasque and CRM (Barcelona) for their kind hospitality in several stages of this work.
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Full metadata record

DC FieldValue Language
dc.contributor.authorBurger, Martinen
dc.contributor.authorCarrillo, Joséen
dc.contributor.authorWolfram, Marie-Thereseen
dc.date.accessioned2016-02-25T12:30:20Zen
dc.date.available2016-02-25T12:30:20Zen
dc.date.issued2010-01en
dc.identifier.citationBurger M, Carrillo J, Wolfram M-T (2010) A mixed finite element method for nonlinear diffusion equations. KRM 3: 59–83. Available: http://dx.doi.org/10.3934/krm.2010.3.59.en
dc.identifier.issn1937-5093en
dc.identifier.doi10.3934/krm.2010.3.59en
dc.identifier.urihttp://hdl.handle.net/10754/597308en
dc.description.abstractWe propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.en
dc.description.sponsorshipJAC was partially supported by MTM2008-06349-C03-03 project DGI-MCI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. MTW acknowledges support by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). We acknowledge the Institute for Pure and Applied Mathematics (University of California, Los Angeles), the International Center for Mathematical Sciences (Edinburgh, UK), the Centro de Ciencias de Benasque and CRM (Barcelona) for their kind hospitality in several stages of this work.en
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)en
dc.subjectMixed finite element methoden
dc.subjectNonlinear diffusion problemsen
dc.subjectOptimal transportation problemen
dc.subjectPatlak-Keller-Segel modelen
dc.subjectPorous medium equationen
dc.titleA mixed finite element method for nonlinear diffusion equationsen
dc.typeArticleen
dc.identifier.journalKinetic and Related Modelsen
dc.contributor.institutionWestfalische Wilhelms-Universitat Munster, Munster, Germanyen
dc.contributor.institutionUniversidad Autonoma de Barcelona, Barcelona, Spainen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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