期刊
KINETIC AND RELATED MODELS
卷 3, 期 1, 页码 59-83出版社
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/krm.2010.3.59
关键词
Nonlinear diffusion problems; optimal transportation problem; mixed finite element method; porous medium equation; Patlak-Keller-Segel model
资金
- DGI-MCI (Spain) [MTM2008-06349-C03-03]
- AGAUR-Generalitat de Catalunya [2009-SGR-345]
- King Abdullah University of Science and Technology (KAUST) [KUK-I1-007-43]
- ICREA Funding Source: Custom
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.
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