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

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 semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.