A mixed finite element method for nonlinear diffusion equations

Martin Burger, José A. Carrillo, Marie Therese Wolfram

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31 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)59-83
JournalKinetic and Related Models
Volume3
DOIs
Publication statusPublished - 1 Mar 2010

Keywords

  • Mixed finite element method
  • Nonlinear diffusion problems
  • Optimal transportation problem
  • Patlak-Keller-Segel model
  • Porous medium equation

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