We introduce a new family of discontinuous Galerkin (DG) finite element schemes for the discretization of first order systems of hyperbolic partial differential equations (PDE) on unstructured simplex meshes in two and three space dimensions that respect the two basic vector calculus identities exactly also at the discrete level, namely that the curl of the gradient is zero and that the divergence of the curl is zero. The key ingredient here is the construction of two compatible discrete nabla operators, a primary one and a dual one, both defined on general unstructured simplex meshes in multiple space dimensions. Our new schemes extend existing cell-centered finite volume methods based on corner fluxes to arbitrary high order of accuracy in space. An important feature of our new method is the fact that only two different discrete function spaces are needed to represent the numerical solution, and the choice of the appropriate function space for each variable is related to the origin and nature of the underlying PDE. The first class of variables is discretized at the aid of a discontinuous Galerkin approach, where the numerical solution is represented via piecewise polynomials of degree N and which are allowed to jump across element interfaces. This set of variables is related to those PDE which are mere consequences of the definitions, derived from some abstract scalar and vector potentials, and for which involutions like the divergence-free or the curl-free property must hold if satisfied by the initial data. The second class of variables is discretized via classical continuous Lagrange finite elements of approximation degree M=N+1 and is related to those PDE which can be derived as the Euler-Lagrange equations of an underlying variational principle.

翻译：本文提出了一类新的间断伽辽金有限元格式，用于在二维和三维空间中的非结构化单纯形网格上离散一阶双曲偏微分方程系统。该格式在离散层面严格保持向量微积分的两个基本恒等式：梯度的旋度为零，以及旋度的散度为零。其核心在于构建两个相容的离散纳布拉算子——一个主算子和一个对偶算子，二者均定义于多维空间中的一般非结构化单纯形网格上。我们的新格式将现有基于角点通量的单元中心有限体积方法推广至任意高阶空间精度。该方法的一个重要特征是仅需两种不同的离散函数空间来表示数值解，且每个变量对应的函数空间选择与偏微分方程的本质和起源相关。第一类变量采用间断伽辽金方法进行离散，其数值解通过分片N次多项式表示，并允许跨单元界面存在跳跃。这类变量对应于那些由抽象标量势和向量势导出的定义性偏微分方程，其初始数据若满足无散或无旋约束，则该约束（即 involutions）必须在整个计算过程中保持。第二类变量采用经典的连续拉格朗日有限元进行离散，其逼近阶为M=N+1。这类变量对应于那些可从变分原理推导出的欧拉-拉格朗日方程形式的偏微分方程。