March  2010, 3(1): 59-83. doi: 10.3934/krm.2010.3.59

A mixed finite element method for nonlinear diffusion equations

1. 

Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität (WWU) Münster, Einsteinstr. 62, D-48149 Münster, Germany

2. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

3. 

DAMTP (Department of Applied Mathematics and Theoretical Physics), University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Received  October 2009 Revised  December 2009 Published  January 2010

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.
Citation: Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
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