发表机构
University of Science and Technology of China; The Hong Kong Polytechnic University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学技术大学; 香港理工大学; 中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于PHT样条的非拟合有限元方法，通过最小化扩展邻域上的L2残差，避免切割子区域构造，实现对称正定系统与最优误差估计，并用后验误差指示器驱动自适应细化。
AI 中文摘要
我们提出了一种新颖的非拟合有限元方法，该方法基于分层T网格上的多项式样条（PHT-splines），用于求解定义在边界由水平集函数隐式描述的域上的椭圆型偏微分方程（PDEs）。利用样条空间增强的光滑性，我们通过在物理域的一个扩展邻域上最小化PDE的$L^2$残差来构建该方法。该公式使得方法能够完全在背景网格上使用标准逐元素求积来实现。它产生一个对称正定线性系统，并避免了边界单元中弯曲切割子区域的几何构造以及相关的非平凡数值积分。我们建立了数值格式的适定性，并在$H^1$范数下推导了最优的先验误差估计。此外，我们采用基于残差的后验误差指示器来指导分层T网格的局部自适应细化。数值实验验证了理论结果，并证明了所提出方法的准确性和效率。
英文摘要
We propose a novel unfitted finite element method based on polynomial splines over hierarchical T-meshes (PHT-splines) for elliptic partial differential equations (PDEs) posed on domains whose boundaries are described implicitly by a level-set function. Exploiting the enhanced smoothness of the spline space, we formulate the method by minimizing the $L^2$ residual of the PDE over an extended neighborhood of the physical domain. This formulation allows the method to be implemented entirely on the background mesh using standard element-wise quadrature. It yields a symmetric positive-definite linear system and avoids the geometric construction of curved cut subregions in boundary elements, as well as the associated nontrivial numerical integration. We establish well-posedness of the numerical scheme and derive optimal \emph{a priori} error estimates in the $H^1$ norm. In addition, we employ a residual-based \emph{a posteriori} error indicator to guide local adaptive refinement of the hierarchical T-meshes. Numerical experiments corroborate the theoretical results and demonstrate the accuracy and efficiency of the proposed approach.
