In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise \(C^2\) smooth interface. It uses piecewise polynomials of degrees k\((k\ge 1)\) and \(k-1\) respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree k for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree \( k-1\) for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in \(L^2\) norm and for the potential approximation in piecewise \(H^1\) seminorm without requiring “sufficiently large” stabilization parameters in the schemes. In addition, error estimation for the potential approximation in \(L^2\) norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.

在本文中，我们针对二维和三维二维二阶椭圆界面问题，提出了两种任意阶扩展的可混合不连续Galerkin（X-HDG）方法。第一种X-HDG方法适用于任何分段\（C ^ 2 \）平滑接口。它分别使用度为k的分段多项式\（（k \ ge 1）\）和\（k-1 \）为子域内部元素内部的势和通量近似，以及度为k的分段多项式用于子域内部元素间边界上的电势的数值轨迹。在元素内部的界面部分采用双值数字轨迹处理跳转条件。第二种X-HDG方法是第一种X-HDG方法的改进版本，适用于任何折线/平面界面，该界面使用度为\（k-1 \）的分段多项式来表示电势的数值轨迹。X-HDG方法具有局部消除特性，因此导致简化的系统，该系统仅涉及元素间边界和界面上势能的数字轨迹的未知数。针对（\ L ^ 2 \）范数中的通量近似和分段\（H ^ 1 \）中的势能近似，得出最佳误差估计。半规范，无需在方案中使用“足够大”的稳定参数。此外，使用对偶参数对\（L ^ 2 \）范数中的近似值进行误差估计。最后，我们提供了几个数值例子来验证理论结果。