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We introduce a new class of Discontinuous Galerkin (DG) methods for solvingnonlinear conservation laws on unstructured Voronoi meshes that use anonconforming Virtual Element basis defined within each polygonal controlvolume. The basis functions are evaluated as an L2 projection of the virtualbasis which remains unknown, along the lines of the Virtual Element Method(VEM). Contrarily to the VEM approach, the new basis functions lead to anonconforming representation of the solution with discontinuous data across theelement boundaries, as typically employed in DG discretizations. To improve thecondition number of the resulting mass matrix, an orthogonalization of the fullbasis is proposed. The discretization in time is carried out following the ADER(Arbitrary order DERivative Riemann problem) methodology, which yields one-stepfully discrete schemes that make use of a coupled space-time representation ofthe numerical solution. The space-time basis functions are constructed as atensor product of the virtual basis in space and a one-dimensional Lagrangenodal basis in time. The resulting space-time stiffness matrix is stabilized byan extension of the dof-dof stabilization technique adopted in the VEMframework, hence allowing an element-local space-time Galerkin finite elementpredictor to be evaluated. The novel methods are referred to as VEM-DG schemes,and they are arbitrarily high order accurate in space and time. The new VEM-DGalgorithms are rigorously validated against a series of benchmarks in thecontext of compressible Euler and Navier-Stokes equations. Numerical resultsare verified with respect to literature reference solutions and compared interms of accuracy and computational efficiency to those obtained using astandard modal DG scheme with Taylor basis functions. An analysis of thecondition number of the mass and space-time stiffness matrix is also forwarded.

Nonconforming Virtual Element basis functions for space-time Discontinuous Galerkin schemes on unstructured Voronoi meshes