Abstract:$L^2$ stable discontinuous Galerkin method with a family of numerical fluxes was studied for the one-dimensional wave equation by Cheng, Chou, Li, and Xing in [Math. Comp. 86 (2017), pp. 121–155]. Although optimal convergence rates were numerically observed with wide choices of parameters in the numerical fluxes, their error estimates were only proved for a sub-family with the construction of a local projection. In this paper, we first complete the one-dimensional analysis by providing optimal error estimates that match all numerical observations in that paper. The key ingredient is to construct an optimal global projection with the characteristic decomposition. We then extend the analysis on optimal error estimate to multidimensions by constructing a global projection on unstructured meshes, which can be considered as a perturbation away from the local projection studied by Cockburn, Gopalakrishnan, and Sayas in [Math. Comp. 79 (2010), pp. 1351–1367] for hybridizable discontinuous Galerkin methods. As a main contribution, we use a novel energy argument to prove the optimal approximation property of the global projection. This technique does not require explicit assembly of the matrix for the perturbed terms and hence can be easily used for unstructured meshes in multidimensions. Finally, numerical tests in two dimensions are provided to validate our analysis is sharp and at least one of the unknowns will degenerate to suboptimal rates if the assumptions are not satisfied.

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中文翻译：

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非结构网格上波动方程的具有广义通量的不连续Galerkin方法的最优误差估计

摘要：程，周，李和邢在[数学。比较 86（2017），第121-155页]。尽管通过数值通量中的各种参数在数值上观察到了最佳收敛速度，但它们的误差估计仅在局部投影构造的子族中得到了证明。在本文中，我们首先通过提供与该论文中所有数值观测值相匹配的最佳误差估计来完成一维分析。关键因素是构造具有特征分解的最佳全局投影。然后，通过在非结构化网格上构建全局投影，将对最佳误差估计的分析扩展到多维，这可以看作是对Cockburn，Gopalakrishnan和Sayas在[Math。比较 79（2010），第1351–1367页]涉及可杂交的不连续Galerkin方法。作为主要贡献，我们使用了新颖的能量论证来证明全局投影的最佳逼近性质。该技术不需要为受扰项显式组装矩阵，因此可以轻松地用于多维的非结构化网格。最后，提供二维数值测试以验证我们的分析是敏锐的，如果不满足假设，则至少一个未知数将退化为次优率。作为主要贡献，我们使用了新颖的能量论证来证明全局投影的最佳逼近性质。该技术不需要为受扰项显式组装矩阵，因此可以轻松地用于多维的非结构化网格。最后，提供二维数值测试以验证我们的分析是敏锐的，如果不满足假设，则至少一个未知数将退化为次优率。作为主要贡献，我们使用了新颖的能量论证来证明全局投影的最佳逼近性质。该技术不需要为受扰项显式组装矩阵，因此可以轻松地用于多维的非结构化网格。最后，提供二维数值测试以验证我们的分析是敏锐的，如果不满足假设，则至少一个未知数将退化为次优率。

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