标题： 用于非拟合域上线性对流扩散方程的扩展杂交间断伽辽金（X-HDG）方法 显示英文标题

标题： eXtended Hybridizable Discontinous Galerkin (X-HDG) Method for Linear Convection-Diffusion Equations on Unfitted Domains

摘要： 在这项工作中，我们提出了一种求解非适配域上线性对流扩散方程（CDE）的新型策略。 在所提出的数值格式中，高阶混合间断Galerkin方法和扩展有限元方法的策略与边界定义的水平集相结合。 该方案因此被称为扩展混合化间断Galerkin（XHDG）方法。 在此方面，混合化间断Galerkin（HDG）方法被扩展到非适配域；即，计算网格不需要拟合到域边界；相反，边界由一个水平集函数定义，并任意穿过背景网格。 保留了HDG的原始未知结构及其确保通量局部守恒的混合性质，同时为被边界切割的单元开发了一个修改后的双线性形式。 在每个被切割的单元上引入边界上的辅助节点迹变量，之后在施加边界条件时消除该变量。 研究了稳态和时间相关的CDE在从扩散到对流主导的各种流动范围内的问题；使用高阶$(p \leq 4)$XHDG通过在任意非适配域上的基准数值例子进行分析。 结果证明，XHDG继承了HDG的最优$(p + 1)$和超优$(p + 2)$收敛特性，同时去除了拟合网格的限制。 显示英文摘要

摘要： 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.