  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
classical polynomials of degree N inside each element, in our new approach the
discrete solution is 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 DG method 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 each sub-triangle
are defined, as usual, on a universal reference element, hence allowing to
compute universal mass, flux and stiffness matrices for the subgrid triangles
once and for all in a pre-processing stage for the reference element only.
Consequently, the construction of an efficient quadrature-free algorithm is
possible, despite the unstructured nature of the computational grid. High order
of accuracy in time is achieved thanks to 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
results have been checked with reference solutions available in literature and
also systematically compared, in terms of computational efficiency and
accuracy, with those obtained by the corresponding modal DG version of the
scheme.
