Polymorphic nodal elements and their application in discontinuous Galerkin methods

In this work we discuss two different but related aspects of the development of efficient discontinuous Galerkin methods on hybrid element grids for the computational modeling of gas dynamics in complex geometries or with adapted grids. In the first part, a recursive construction of different nodal sets for hp finite elements is presented. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. They share the property that the nodes along the sides of the two-dimensional elements and along the edges of the three-dimensional elements are the Legendre-Gauss-Lobatto points. In the second part, we apply these nodal elements as the basis for discontinuous Galerkin schemes. We shall discuss both the immediate nodal formulation as well as the widely used modal formulation where in the latter case a nodal based integration technique is introduced to reduce the computational cost. We shall illustrate the performance of the scheme on several large scale applications and discuss its use in a recently developed space-time expansion discontinuous Galerkin scheme.