Error estimates for numerical approximations of a nonlinear gradient flow model
This work offers rigorous numerical analysis for a class of nonlinear gradient flows, benefiting researchers in numerical PDEs and image processing, but is incremental as it extends existing GDM theory to a specific model.
The paper provides error estimates for a fully discretized implicit scheme of a nonlinear gradient flow model using the gradient discretisation method, proving existence, uniqueness, stability, and consistency, with numerical results from conforming and nonconforming finite elements.
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.