A novel hybrid difference method for an elliptic equation

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.