Local Virtual Element Basis Functions for Space-Time Discontinuous Galerkin Schemes on Unstructured Voronoi Meshes (Communications in Computational Physics, 2024)
We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a local Virtual Element basis defined within each polygonal control volume. The basis functionsare evaluated as an $L_2$ projection of the virtual basis which remains unknown, alongthe lines of the Virtual Element Method (VEM). Contrary to the VEM approach, thenew basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as employed in DG discretizations.The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes thatmake use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the novel local virtual basisin space and a one-dimensional Lagrange nodal basis in time. The resulting space-time stiffness matrix is stabilized by an extension of the dof–dof stabilization technique adopted in the VEM framework, hence allowing an element-local space-timeGalerkin finite element predictor to be evaluated. The new VEM-DG algorithms arerigorously validated against a series of benchmarks in the context of compressible Euler and Navier–Stokes equations.
Keywords: Discontinuous Galerkin; Virtual Element Method; high order in space and time; ADER schemes; unstructured meshes.