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A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planckequations in space dimensions $d\ge2$ is presented. It applies to a large classof nonlinear diffusion equations, whose dynamics are driven by internalenergies and given external potentials, e.g. the porous medium equation and thefast diffusion equation. The key ingredient in our approach is the gradientflow structure of the dynamics. For discretization of the Lagrangian map, weuse a finite subspace of linear maps in space and a variational form of theimplicit Euler method in time. Thanks to that time discretisation, the fullydiscrete 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 numericalexperiments for the porous medium equation indicates that the scheme iswell-adapted to track the growth of the solution's support.

A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes