In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method is combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e, the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order $(p \leq 4)$ XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal $(p + 1)$ and super $(p + 2)$ convergence properties of HDG while removing the fitting mesh restriction.

翻译：本文针对非拟合域上的线性对流扩散方程（CDE）数值求解提出了一种新策略。在所提出的数值格式中，高阶混合化不连续伽辽金方法（HDG）与扩展有限元方法（XFEM）的策略相结合，并采用水平集函数定义边界。因此，该方法被命名为扩展混合化不连续伽辽金（X-HDG）方法。在此框架下，混合化不连续伽辽金（HDG）方法被扩展至非拟合域；即计算网格无需贴合域边界，而是通过水平集函数定义边界，使其任意切割背景网格。在保持HDG原有未知量结构及其混合特性（确保通量局部守恒）的同时，针对边界切割单元开发了修正的双线性形式。在每个切割单元中，引入边界上的辅助节点迹变量，该变量在施加边界条件后被消去。通过任意非拟合域上的基准数值算例，采用高阶（p ≤ 4）X-HDG研究了从扩散主导到对流主导的多种流态下的稳态和时变对流扩散方程。结果表明，X-HDG继承了HDG的最优（p + 1）和超收敛（p + 2）性质，同时消除了网格拟合约束。