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.

翻译：我们针对非结构化Voronoi网格上的非线性守恒律提出了一类新的间断Galerkin（DG）方法，该方法在每个多边形控制体内采用基于非协调虚拟元（Virtual Element）基函数的表示。这些基函数通过未知虚拟基的L2投影进行评估，其思路与虚拟元方法（VEM）一致。与VEM方法不同，新基函数导致解的表示具有非协调性，即在单元边界上数据不连续，这恰是DG离散化的典型特征。为改善所得质量矩阵的条件数，本文提出了对完整基函数进行正交化的方案。时间离散采用ADER（任意阶导数黎曼问题）方法，通过耦合时空数值解的表示构建一步全离散格式。时空基函数由空间虚拟基与时间一维拉格朗日节点基的张量积构造而成。通过扩展VEM框架中的自由度-自由度（dof-dof）稳定技术，对所得时空刚度矩阵进行稳定化处理，从而允许在单元局部进行时空Galerkin有限元预测器的评估。新方法称为VEM-DG格式，其在空间和时间上均具有任意高阶精度。针对可压缩欧拉和纳维-斯托克斯方程的一系列基准测试，我们对新VEM-DG算法进行了严格验证。数值结果与文献参考解进行了对比，并在精度和计算效率上与采用泰勒基函数的标准模态DG方案进行了比较。此外，本文还分析了质量矩阵和时空刚度矩阵的条件数。