We study a mixed discontinuous Galerkin (DG) method for solving the second-order wave equation. The stress variable $\mathit{p}$ and the displacement variable $u$ are discretized by the mixed DG element pair ${\mathit{P}}_{k+1}$–$P_k$ ($k\ge 0$). Under appropriate regularity assumptions on the solution pair, we derive optimal error estimates for the spatially semi-discrete scheme. Specifically, we prove the optimal convergence orders for $u$ in $L^2$ norm and $\mathit{p}$ in ${\mathit{L}}^2$ norm. Some test problems are presented, and the numerical results show that the orders of convergence coincide with the predicted error estimates.

中文翻译：

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波动方程的混合不连续Galerkin方法

我们研究了混合不连续伽勒金（DG）方法来求解二阶波动方程。应力变量$\mathit{p}$ 和位移变量 $ü$ 被混合的DG元素对离散化 ${\mathit{P}}_{ķ+\mathrm{1个}}$–$P_{ķ}$ （$ķ\ge 0$）。在解对上适当的正则性假设下，我们导出空间半离散方案的最佳误差估计。具体来说，我们证明了最优的收敛阶$ü$ 在 ${\mathrm{大号}}^2$ 规范和 $\mathit{p}$ 在 ${\mathit{大号}}^2$规范。提出了一些测试问题，数值结果表明收敛阶次与预测误差估计相符。