Error estimates for numerical approximations of a nonlinear gradient flow model

TL;DR

This paper analyzes the error estimates of a fully discretized implicit scheme for a nonlinear gradient flow model, using the gradient discretisation method, and provides numerical validation with finite element methods.

Contribution

It introduces a comprehensive error analysis framework for a nonlinear gradient flow model using GDM, including existence, stability, and convergence results.

Findings

Proved existence and uniqueness of the scheme solution.

Established stability and consistency of the discretization.

Provided numerical results validating the theoretical error estimates.

Abstract

We perform numerical analysis of a nonlinear gradient flow, which can be regarded as a parabolic minimal surface problem or a regularised total variation flow, using the gradient discretisation method (GDM). GDM is a unified convergence analysis framework that covers conforming and nonconforming numerical methods, for instance, conforming and nonconforming finite element, two-point flux approximation, etc.. In this paper, a fully discretised implicit scheme of the model is proposed, the existence and uniqueness of the solution to the scheme is proved, the stability and consistency of the scheme are analysed, and error estimates are established. Numerical results based on the conforming and nonconforming $\mathbb{P}^1$ finite elements are also provided.