Title: A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes,
Abstract: A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck
equations in space dimensions $d\ge2$ is presented. It applies to a large class
of nonlinear diffusion equations, whose dynamics are driven by internal
energies and given external potentials, e.g. the porous medium equation and the
fast diffusion equation. The key ingredient in our approach is the gradient
flow structure of the dynamics. For discretization of the Lagrangian map, we
use a finite subspace of linear maps in space and a variational form of the
implicit Euler method in time. Thanks to that time discretisation, the fully
discrete solution inherits energy estimates from the original gradient flow,
and these lead to weak compactness of the trajectories in the continuous limit.
Consistency is analyzed in the planar situation, $d=2$. A variety of numerical
experiments for the porous medium equation indicates that the scheme is
well-adapted to track the growth of the solution's support.