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 language | English (US) |
|---|---|
| Pages (from-to) | 59-83 |
| Number of pages | 25 |
| Journal | Kinetic and Related Models |
| Volume | 3 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 21 2010 |
| Externally published | Yes |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.