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. Introduction.The study of the evolution of transition interfaces, which is of paramount importance in many physical/biological phenomena and industrial processes, can be grouped into two macro classes, each one corresponding to a different method of dealing with the moving free-boundary: the sharp interface method and the phase-field method. In the sharp interface approach, the free boundary is to be determined together with the solution of suitable partial differential equations where proper jump relations have to imposed across the free boundary. In the phase field approach, the interface is specified as the level set of a smooth continuos function exhibiting large gradients across the interface.Phase field models, which date back to the works of Korteweg [33], Cahn and Hilliard [13,30,31], Landau and Ginzburg [34] and van der Waals [43], have been classicaly employed to describe phase separation in binary alloys. However, recently Cahn-Hilliard type equations have been extensively used in an impressive variety of applied problems, such as, among the others, tumor growth [47,39], origin of Saturn's rings [42], separation of di-block copolymers [15], population dynamics [17], image processing [9] and even clustering of mussels [35].Due to the wide spectrum of applications, the study of efficient numerical methods for the approximate solution of the Cahn-Hilliard equation has been the object of an intensive research activity. Summarizing the achievements in this field is a tremendous task that go beyond the scope of this paper. Here, we limit ourvselves to some remarks on finite element based methods, as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. As the Cahn-Hilliard equation is a fourth order nonlinear problem, a natural approach is the use of C 1 finite elements (FEM) as in [25,21]. However, in order to avoid the well known difficulty met in the implementation of C 1 finite elements, another possibility is the use of non-conforming (see, e.g., [22]) or discontinuous (see, e.g., [46]) methods; the drawback is that in such case the discrete solution will not satisfy a C 1 regularity. Alternatively, the most common strategy employed in practice to solve the Cahn-Hilliard equation with (continuos and discontinuous) finite elements is 2 to use mixed methods (see e.g. [23, 24] and [32] for the continuous and discontinuous setting, respecti...

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