A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes

Abstract

In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn-Hilliard equation. The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semi-discrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.

References in corpus (1)

• A Hybrid High-Order method for the incompressible Navier--Stokes equations based on Temam's device

Cited by in corpus (1)

• Virtual Element Methods for hyperbolic problems on polygonal meshes