$C^1$-VEM for some variants of the Cahn-Hilliard equation: a numerical exploration

TL;DR

This paper explores the application of the $C^1$-Virtual Element Method to approximate variants of the Cahn-Hilliard equation on polygonal meshes, including advective and inpainting problems, with numerical results demonstrating effectiveness.

Contribution

It introduces a $C^1$-VEM approach for complex Cahn-Hilliard variants on polygonal meshes, expanding numerical methods for these problems.

Findings

Effective numerical approximation demonstrated.

Method applicable to advective and inpainting variants.

Numerical results confirm method's efficacy.

Abstract

We consider the $C^1$-Virtual Element Method (VEM) for the conforming numerical approximation of some variants of the Cahn-Hilliard equation on polygonal meshes. In particular, we focus on the discretization of the advective Cahn-Hilliard problem and the Cahn-Hilliard inpainting problem. We present the numerical approximation and several numerical results to assess the efficacy of the proposed methodology.