Abstract This paper proposes a discontinuous Galerkin method to solve the generalized Sobolev equation. In this numerical procedure, the temporal variable has been discretized by the Crank–Nicolson idea to get a time-discrete scheme with the second-order accuracy. Then, in the second stage the spatial variable has been discretized by the discontinuous Galerkin finite element method. A prior error estimate has been proposed for the semi-discrete scheme based on the spatial discretization. By applying the Crank–Nicolson idea a full-discrete scheme is driven. Furthermore, an error estimate has been proved to get the convergence order of the developed scheme. Finally, some numerical examples have been presented to show the efficiency and theoretical results of the new numerical procedure.

中文翻译：

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求解广义Sobolev方程的内罚不连续Galerkin技术

摘要 本文提出了一种求解广义Sobolev方程的间断Galerkin方法。在这个数值过程中，时间变量已经被 Crank-Nicolson 思想离散化，以获得具有二阶精度的时间离散方案。然后，在第二阶段，空间变量已经被非连续伽辽金有限元方法离散化。已经为基于空间离散化的半离散方案提出了先验误差估计。通过应用 Crank-Nicolson 思想，可以驱动一个完全离散的方案。此外，已证明误差估计可以得到所开发方案的收敛阶次。最后，给出了一些数值例子来说明新数值程序的效率和理论结果。