In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is $L^2$-stable and provides the optimal $(p+1)$-th order of accuracy for smooth solutions when using piecewise p-th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order $p+1$ toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise $P^p$ polynomials with arbitrary $p\ge 1$. Several numerical results are provided to confirm the convergence of the proposed scheme.

在本文中，我们为非线性三阶常微分方程提出了一种最佳收敛的不连续伽勒金（DG）方法。建立了该解和两个辅助变量的收敛性质，它们近似于该解的一阶和二阶导数。更具体地说，我们证明该方法是${\mathrm{大号}}^2$稳定并提供最佳 $（p+\mathrm{1个}）$使用分段p次多项式时的光滑解的3阶精度。此外，我们证明DG解的导数是有序超闭合的$p+\mathrm{1个}$朝向精确解的高斯-拉多投影的导数。该证明适用于任意非均匀规则网格和分段$P^p$ 任意多项式 $p\ge \mathrm{1个}$。提供了一些数值结果，以确认所提出方案的收敛性。