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.

中文翻译：

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退化抛物线方程能量稳定有限元逼近的收敛性和后验误差分析

我们以非线性各向异性 Fokker-Planck 方程的形式为退化抛物线问题的数值逼近提出了一种有限元方案。该方案是能量稳定的，在其定义中仅涉及物理激励量，并且能够处理一般的非结构化网格。由于紧凑性论证，在非常一般的假设下，它的收敛性得到了严格证明。尽管该方案基于 1 次拉格朗日有限元，但在局部后处理后产生平衡通量后，它是局部保守的。这也允许导出近似解的有保证的后验误差估计。给出了数值实验，以证明所提出的方案在涉及强各向异性和漂移项的各种情况下的良好行为。