A higher order nonconforming virtual element method for the Cahn-Hilliard equation

TL;DR

This paper introduces a nonconforming virtual element method of arbitrary order for the 2D Cahn-Hilliard equation, providing error analysis and numerical verification of convergence.

Contribution

It develops a fully nonconforming VEM for the Cahn-Hilliard equation with error analysis and a fully discrete scheme using convex splitting Runge-Kutta methods.

Findings

The method achieves theoretical convergence as verified numerically.

Error estimates are established for the semidiscrete scheme.

Numerical experiments confirm the effectiveness of the fully discrete scheme.

Abstract

In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.