Discontinuous Galerkin method with arbitrary polygonal finite elements

<p>In the paper the efficient application of discontinuous Galerkin (DG) method on polygonal meshes is presented. Three versions of DG method are under consideration in which the approximation is constructed using sets of arbitrary basis functions. It means that in the presented approach there is no need to define nodes or to construct shape functions. The shape of a polygonal finite element (FE) can be quite arbitrary. The polygonal finite (FE) element can be of quite arbitrary shape. 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 limited to two-dimensional case. However, direct extension of the algorithms to three-dimension is possible.