A mixed finite element method for nonlinear diffusion equations
The solution of a class of nonlinear diffusion equations is considered, which includes, e.g., the heat transfer equation, the porous media equation, the fast diffusion equation, and the Patlak-Keller-Segel model. Based on an optimal transport formulation of the nonlinear diffusion equation a numerical solution method is proposed. At first a linearization of the problem in optimal transport formulation and a time discretization are presented. For the spatial discretization a mixed finite element method is used. A maximum principle for the semi-implicit scheme and the fully discrete scheme is proved. Additionally, it is shown that the semi-implicit scheme preserves the exponential decay of the relative energy functional. The existence and uniqueness of the solution of the fully discrete problem is proved. The application of the proposed semi-implicit scheme to the porous media equation and to the fast diffusion equation is discussed. The behaviour of the solution obtained by means of the proposed method is illustrated by computational experiments. Finally, the application of the presented scheme to the Patlak-Keller-Segel model and corresponding numerical experiments are described.
scientific article; zbMATH DE number 5705213