Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes (Communications in Computational Physics, 2022)
We propose a new high order accurate nodal discontinuous Galerkin (DG)method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classicalpolynomials of degree $N$ inside each element, in our new approach the discrete solutionis represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DGmethod on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on eachsub-triangle are defined, as usual, on a universal reference element, hence allowing tocompute universal mass, flux and stiffness matrices for the subgrid triangles once andfor all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructurednature of the computational grid. High order of accuracy in time is achieved thanksto the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical resultshave been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtainedby the corresponding modal DG version of the scheme.