In this article, we discuss the numerical solution for the two‐dimensional (2‐D) damped sine‐Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher‐order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum‐norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.

中文翻译：

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正弦阻尼方程的连续Galerkin方法数值分析

在本文中，我们将讨论使用时空连续Galerkin方法求解二维（2-D）阻尼正弦-戈登方程的数值解。该方法具有可变的时间步长和空间网格结构，其离散方案具有良好的稳定性，这对于非结构化网格的自适应计算是必需的。同时，它可以轻松地在时空两个方向上获得较高的精度。严格证明了数值解的存在性和唯一性，并且给出了最大范数的先验误差估计，而没有附加任何时空网格条件。同样，我们证明了，如果以合理的方式生成每个时间级别的网格，我们可以在时间和空间变量上获得最优的收敛顺序。最后，