Unconditionally energy stable C0-virtual element scheme for solving generalized Swift-Hohenberg equation

Abstract

The very recently introduced Virtual Element Method (VEM) is a numerical method for solving partial differential equations that was created out of the mimetic difference method, but was later reformulated into the Galerkin framework. It is a generalization of the standard Finite Element Method (FEM) to general meshes made up by arbitrary polyhedra. The greatest advantage of VEM is its ability to deal with very complex geometries, i.e., made up by elements of any number of edges not necessarily convex, hanging nodes, flat angles, collapsing nodes, etc., while retaining the same approximation properties of FEM. In this article, the C0-virtual element method is formulated and analyzed to solve generalized Swift-Hohenberg equation on polygonal meshes. The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here, we present the spatial VE discretization based on a mixed formulation for the nonlinear Swift-Hohenberg equation as a class of fourth-order gradient flow problems. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second-order in time. By following the algebraic implementation of the discrete system, we provide numerical tests validating the theoretical estimates and plotting two-dimensional pattern formation problems.