Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations. (English) Zbl 1456.65105
Summary: We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K65 | Degenerate parabolic equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35Q84 | Fokker-Planck equations |
Keywords:
degenerate parabolic equation; Fokker-Planck equation; energy-stable discretization; local conservation; equilibrated flux; convergence; a posteriori error estimate.Software:
FreeFem++References:
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