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We propose a new high order accurate nodal discontinuous Galerkin (DG) methodfor the solution of nonlinear hyperbolic systems of partial differentialequations (PDE) on unstructured polygonal Voronoi meshes. Rather than usingclassical polynomials of degree N inside each element, in our new approach thediscrete solution is represented by piecewise continuous polynomials of degreeN within each Voronoi element, using a continuous finite element basis definedon a subgrid inside each polygon. We call the resulting subgrid basis anagglomerated finite element (AFE) basis for the DG method on general polygons,since it is obtained by the agglomeration of the finite element basis functionsassociated with the subgrid triangles. The basis functions on each sub-triangleare defined, as usual, on a universal reference element, hence allowing tocompute universal mass, flux and stiffness matrices for the subgrid trianglesonce and for all in a pre-processing stage for the reference element only.Consequently, the construction of an efficient quadrature-free algorithm ispossible, despite the unstructured nature of the computational grid. High orderof accuracy in time is achieved thanks to the ADER approach, making use of anelement-local space-time Galerkin finite element predictor.  The novel schemes are carefully validated against a set of typical benchmarkproblems for the compressible Euler and Navier-Stokes equations. The numericalresults have been checked with reference solutions available in literature andalso systematically compared, in terms of computational efficiency andaccuracy, with those obtained by the corresponding modal DG version of thescheme.

Continuous finite element subgrid basis functions for Discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes