We prove an abstract convergence result for a family of dual-mesh based quadrature rules on tensor products of simplical meshes. In the context of the multilinear tensor-product finite element discretization of reaction-drift-diffusion equations, our quadrature rule generalizes the mass-lump rule, retaining its most useful properties; for a nonnegative reaction coefficient, it gives an $O(h^2)$-accurate, nonnegative diagonalization of the reaction operator. The major advantage of our scheme in comparison with the standard mass lumping scheme is that, under mild conditions, it produces an $O(h^2)$ consistency error even when the integrand has a jump discontinuity. The finite-volume-type quadrature rule has been stated in a less general form and applied to systems of reaction-diffusion equations related to particle-based stochastic reaction-diffusion simulations (PBSRD); in this context, the reaction operator is \textit{required} to be an $M$-matrix and a standard model for bimolecular reactions has a discontinuous reaction coefficient. We apply our convergence results to a finite element discretization of scalar drift-diffusion-reaction model problem related to PBSRD systems, and provide new numerical convergence studies confirming the theory.

翻译：我们证明了在单纯形网格张量积上基于对偶网格的一类求积法则的抽象收敛性结果。在反应-漂移-扩散方程的多线性张量积有限元离散化背景下，我们的求积法则推广了质量集中法则，并保留了其最有用的性质：对于非负反应系数，它给出了反应算子的$O(h^2)$精度、非负对角化。与标准质量集中格式相比，我们方案的主要优势在于，在温和条件下，即使被积函数存在跳跃间断，它也能产生$O(h^2)$阶的一致性误差。该有限体积型求积法则曾以较不一般的形式提出，并应用于与基于粒子的随机反应-扩散模拟（PBSRD）相关的反应-扩散方程组；在此背景下，反应算子被\textit{要求}为$M$-矩阵，且双分子反应的标准模型具有间断反应系数。我们将收敛性结果应用于与PBSRD系统相关的标量漂移-扩散-反应模型问题的有限元离散化，并提供了新的数值收敛研究以验证理论。