In this paper, we derive two a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to linear second-order elliptic problems on Cartesian grids. We first prove that the gradient of the LDG solution is superconvergent with order \(p+1\) towards the gradient of Gauss-Radau projection of the exact solution, when tensor product polynomials of degree at most p are used. Then, we prove that the gradient of the actual error can be split into two parts. The components of the significant part can be given in terms of \((p+1)\)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We further develop a postprocessing gradient recovery scheme for the LDG solution. This recovered gradient superconverges to the gradient of the true solution. The order of convergence is proved to be \(p+1\). We use our gradient recovery result to develop a robust recovery-type a posteriori error estimator for the gradient approximation which is based on an enhanced recovery technique. We prove that the proposed residual-type and recovery-type a posteriori error estimates converge to the true errors in the \(L^2\)-norm under mesh refinement. The order of convergence is proved to be \(p + 1\). Moreover, the proposed estimators are proved to be asymptotically exact. Finally, we present a local adaptive mesh refinement procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for \(P^p\) polynomials with \(p\ge 1\). We provide several numerical examples illustrating the effectiveness of our procedures.

在本文中，我们导出了笛卡尔网格上线性二阶椭圆问题的局部不连续Galerkin（LDG）方法的两个后验误差估计。我们首先证明，当使用度数最多为p的张量积多项式时，LDG解的梯度朝着精确解的Gauss-Radau投影的梯度以\（p + 1 \）阶超收敛。然后，我们证明了实际误差的梯度可以分为两部分。有效部分的成分可以根据（（p + 1）\）- Radau多项式给出。我们使用这些结果来构建可靠且有效的残差型后验误差估计。我们进一步为LDG解决方案开发了一种后处理梯度恢复方案。该恢复的梯度超收敛到真实解的梯度。收敛的阶数被证明为\（p + 1 \）。我们使用梯度恢复结果来开发一种基于增强恢复技术的鲁棒恢复型后验误差估计器，用于梯度近似。我们证明了在网格细化下，提出的残差型和恢复型后验误差估计收敛到\（L ^ 2 \）-范数中的真实误差。证明收敛的阶数是\（p + 1 \）。此外，证明了所提出的估计量是渐近精确的。最后，我们提出了一种局部自适应网格细化程序，该程序利用了我们的局部和全局后验误差估计。我们的证明对于任意规则网格和\（p \ ge 1 \）的\（P ^ p \）多项式都是有效的。我们提供了几个数值示例来说明我们的程序的有效性。