Hybrid difference methods are a kind of finite difference methods which is similar to hybrid discontinuous Galerkin methods introduced by Jeon and Park (SIAM J. Numer. Anal., 2010). In the previous hybrid difference method, the approximate solution is only defined on the lines parallel to the coordinate axes, but the approximation is not defined at the corner/edge nodes. Thus, it is hard to calculate the approximation value without loosing accuracy at any point in the domain due to the aforementioned missing information. To overcome this issue, we first propose a novel hybrid difference method by imposing some conditions to determine the missing information. With this, we are able to provide not only continuous approximations in the whole domain, but also the gradient of the numerical solution becomes continuous under certain conditions. Next, we prove that the proposed method is stable and locally conservative. The proposed method can be also viewed as the existing hybrid difference method with a simple postprocessing. The postprocessing not only induces a simple tridiagonal system on each mesh line, but also it can be done efficiently line by line or in parallel. Lastly, a reliable and efficient a posteriori error estimate is established for computational efficiency. Several numerical results are presented to confirm our findings.

混合差分法是一种类似于 Jeon 和 Park (SIAM J. Numer. Anal., 2010) 引入的混合不连续伽辽金方法的有限差分方法。在之前的混合差分法中，近似解只定义在平行于坐标轴的直线上，而在角/边节点处没有定义近似。因此，由于上述缺失信息，很难在不损失域中任何点的精度的情况下计算近似值。为了克服这个问题，我们首先提出了一种新的混合差分方法，通过施加一些条件来确定缺失的信息。有了这个，我们不仅可以在整个域中提供连续的近似值，而且数值解的梯度在某些条件下变得连续。下一个，我们证明了所提出的方法是稳定的和局部保守的。所提出的方法也可以看作是具有简单后处理的现有混合差分方法。后处理不仅在每条网格线上引入一个简单的三对角系统，而且可以逐行或并行有效地完成。最后，为计算效率建立了可靠且有效的后验误差估计。提供了几个数值结果来证实我们的发现。为计算效率建立了可靠且有效的后验误差估计。提供了几个数值结果来证实我们的发现。为计算效率建立了可靠且有效的后验误差估计。提供了几个数值结果来证实我们的发现。