Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure

TL;DR

This paper introduces a finite volume numerical scheme for degenerate parabolic equations with gradient flow structure, ensuring discrete energy dissipation, convergence, and robustness, supported by theoretical proofs and numerical experiments.

Contribution

The paper develops a novel finite volume scheme that preserves the gradient flow structure at the discrete level for complex parabolic equations.

Findings

The scheme guarantees energy diminishing behavior numerically.

Convergence of the scheme is proven under broad conditions.

Numerical tests demonstrate robustness and efficiency.

Abstract

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach.