EnVarA-FEM for the flux-limited porous medium equation

Abstract

The flux-limited porous medium equation is a kind of nonlinear degenerate parabolic equation based on the porous medium equation, its solution not only has a finite propagation speed and possible waiting time, but also may develop jump fronts. It is difficult to construct numerical methods to capture these features in an effective way. We propose structure-preserving finite element numerical schemes based on the energetic variational approach, which are proved to be uniquely solvable, positivity preserving and energy dissipative. The analysis of the free boundaries conditions is given to illustrate the circumstances under which the solution of the flux-limited porous medium equation will have finite propagation speed, waiting time or jump fronts. Moreover, the optimal convergence analysis for the nonlinear numerical scheme is presented by applying the technique of combining the rough and refined error estimates. Numerical experiments reveal that the proposed numerical method can successfully capture important features of the flux-limited PME: finite propagation speed of the free boundary, waiting time, effect of amplitudes on the regularity of the solutions and jump fronts. In particular, the waiting time is computed naturally and the convergence order of the waiting time can be given numerically. A criterion to approximate the time where discontinuous interfaces occur is also established, and the corresponding convergence order is given numerically.