Discontinuous Galerkin method with arbitrary polygonal finite elements

Abstract

The paper presents the efficient application of discontinuous Galerkin (DG) method on polygonal meshes. Three versions of the DG method in which the approximation is constructed using sets of arbitrary basis functions are under consideration. The analysed approach does not require definition of nodes or construction of shape functions. The shape of a polygonal finite element (FE) can be quite arbitrary. It can have arbitrary number of edges and can be non-convex. In particular, a single FE can have a polygonal hole or can even consist of two or more completely separated parts. The efficiency, flexibility and versatility of the presented approach is illustrated with a set of benchmark examples. The paper is restricted to two-dimensional case. However, direct extension of the algorithms to three-dimensions is possible.