发表机构
Sichuan University; Virginia Tech; Oklahoma State University（四川大学; 弗吉尼亚理工大学; 俄克拉荷马州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究开发了适用于曲域二维椭圆边值与界面问题的任意阶GC-FE框架，耦合GC-FE与GC-IFE空间，避免几何变分犯罪，数值实验验证其精度与收敛阶。
AI 中文摘要
我们针对曲域上的二维椭圆边值问题和界面问题，开发了任意阶的贴合几何有限元（GC-FE）框架。利用Frenet-Serret变换，曲边界和贴合界面的线段可被精确表示，而Frenet坐标下的多项式会在物理坐标中生成非多项式的局部形函数。对于非贴合界面的网格，曲边界单元上的GC-FE空间与界面切割单元上的贴合几何浸入式有限元（GC-IFE）空间耦合，其余区域采用标准多项式空间。我们为GC-FE空间建立了最优逼近、逆和迹估计。对于贴合网格，我们证明了对称内罚间断Galerkin离散的适定性，以及能量范数和L²范数下的最优误差估计。通过精确保留给定曲线，该方法避免了与曲几何逼近相关的几何变分犯罪，无需对应的几何一致性估计。数值实验验证了预测的收敛阶，显示其全局精度与节点等参有限元相当，且在报告的测试中具有更小的真实界面迹误差，并在非贴合界面的网格上验证了耦合的GC-FE-GC-IFE方法。
英文摘要
We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet--Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate generally nonpolynomial local shape functions in physical coordinates. For interface-unfitted meshes, GC-FE spaces on curved-boundary elements are coupled with geometry-conforming immersed finite element (GC-IFE) spaces on interface-cut elements, with standard polynomial spaces used elsewhere. We establish optimal approximation, inverse, and trace estimates for the GC-FE spaces. For fitted meshes, we prove well-posedness and optimal error estimates in energy and $L^2$ norms for a symmetric interior penalty discontinuous Galerkin discretization. By retaining the prescribed curves exactly, the method avoids the geometric variational crime associated with curved-geometry approximation and requires no corresponding geometric consistency estimates. Numerical experiments confirm the predicted rates, show global accuracy comparable to nodal isoparametric finite elements and smaller true-interface trace errors in the reported tests, and demonstrate the coupled GC-FE-GC-IFE method on interface-unfitted meshes.
