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

TL;DR

The paper addresses the challenge of achieving high-order accuracy for nonlinear conservation laws on unstructured polygonal meshes by marrying the Virtual Element Method with discontinuous Galerkin discretizations. It develops nonconforming VEM-DG schemes in space-time using ADER time stepping, leveraging an $L^2$ projection-based spatial basis and a tensor-product space-time basis to obtain fully discrete one-step methods. A Gram–Schmidt-type orthogonalization and a dof-dof stabilization are introduced to mitigate ill-conditioning of VEM matrices, enabling stable, high-order performance. Across compressible Euler and Navier–Stokes benchmarks, the proposed methods show accurate convergence, robust shock-capturing with limited artificial viscosity, and competitive efficiency compared with modal DG and AFE-DG schemes, highlighting the flexibility and potential for quadrature-free extensions on general meshes.

Abstract

We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a nonconforming Virtual Element basis defined within each polygonal control volume. The basis functions are evaluated as an L2 projection of the virtual basis which remains unknown, along the lines of the Virtual Element Method (VEM). Contrarily to the VEM approach, the new basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as typically employed in DG discretizations. To improve the condition number of the resulting mass matrix, an orthogonalization of the full basis is proposed. The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes that make use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the virtual basis in 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-time Galerkin finite element predictor 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-DG algorithms are rigorously validated against a series of benchmarks in the context of compressible Euler and Navier-Stokes equations. Numerical results are verified with respect to literature reference solutions and compared in terms of accuracy and computational efficiency to those obtained using a standard modal DG scheme with Taylor basis functions. An analysis of the condition number of the mass and space-time stiffness matrix is also forwarded.

Figures (9)

• Figure 1: Comparison between VEM-DG (squares), M-DG (circles) and AFE-DG (diamonds) schemes from second up to fourth order of accuracy. Left: dependency of the error norm on the mesh size. Right: dependency of the error norm on the CPU time.
• Figure 2: $L_2$ projection of the density distribution of the initial condition for the isentropic vortex test case with $N=2$ (top row) and $N=3$ (bottom row) on a mesh with characteristic size $h_{\Omega}=10/12$. Left: VEM-DG. Middle: M-DG. Right: AFE-DG.
• Figure 3: Analysis of the condition number of the VEM matrices employed in the new VEM-DG schemes for $N=1$ (top row), $N=2$ (middle row) and $N=3$ (bottom row) for a mesh with characteristic size $h_{\Omega}=1/3$. Left column: logarithm of the condition number of the space mass matrix $\mathbf{M}$. Right column: logarithm of the condition number of the space-time stiffness matrix $\mathbf{K}_1$.
• Figure 4: First problem of Stokes at time $t_f=1$. Third order numerical results for the vertical component of the velocity obtained with the VEM-DG scheme and compared against the reference solution by extracting a one-dimensional cut of 200 equidistant points along the $x-$direction at $y=0$. Viscosity $\mu=10^{-3}$ (left) and $\mu=10^{-4}$ (right).
• Figure 5: Explosion problem at time $t_f=0.25$. Third order numerical results with VEM-DG scheme for density, horizontal velocity and pressure compared against the reference solution extracted with a one-dimensional cut of 200 equidistant points along the $x-$direction at $y=0$. Three-dimensional view of the density distribution is shown in the top left panel.

Theorems & Definitions (4)

• Remark : Orthogonalization of the Virtual Element basis
• Remark : Derivatives of the Virtual Element basis
• Remark : Computation of the space-time stiffness matrices
• Remark : Computation of the mass matrix