Polymorphic nodal elements and their application in discontinuous Galerkin methods

Abstract

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. 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. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. In the second part, we apply these nodal elements as the basis for discontinuous Galerkin schemes. We use a modal based formulation and introduce a nodal based integration technique to reduce computational cost. We 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.

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

“…In [18], the definition of the Lagrangian basis functions in the physical frame is strictly related to the number of nodes defining the geometry of the element. The approach has been extended to dG methods by Gassner, Lörcher, Munz and Hesthaven [8] in order to obtain a quadrature free approach.…”

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

“…Gassner et.al. [10] used polymorphic nodal element in the modal based formulation to reduce the cost of numerical integrations. However one obvious shortcoming of these formulations is the high computational cost of the surface and volume integrations coming from the weighted residual formulation.…”

Simulation of the Taylor–Green Vortex Using High-Order Flux Reconstruction Schemes

“…The DG Spectral Element Method (DGSEM) represents the solution in classical form but performs integration cheaply using cheap quadrature formulae [8,9]. Gassner et al [10] devised efficient quadrature schemes for arbitrarily shaped elements based on the nodal DG approach.…”