Recent advancements in finite element methods allows for the implementation of mesh cells with curved edges. In the present work, we develop the tools necessary to employ multiply connected mesh cells, i.e. cells with holes, in planar domains. Our focus is efficient evaluation the $H^1$ semi-inner product and $L^2$ inner product of implicitly-defined finite element functions of the type arising in boundary element based finite element methods (BEM-FEM) and virtual element methods (VEM). These functions may be defined by specifying a polynomial Laplacian and a continuous Dirichlet trace. We demonstrate that these volumetric integrals can be reduced to integrals along the boundaries of mesh cells, thereby avoiding the need to perform any computations in cell interiors. The dominating cost of this reduction is solving a relatively small Nystrom system to obtain a Dirichlet-to-Neumann map, as well as the solution of two more Nystrom systems to obtain an ``anti-Laplacian'' of a harmonic function, which is used for computing the $L^2$ inner product. We demonstrate that high-order accuracy can be achieved with several numerical examples.
High-Order Quadrature for Implicitly Defined Surfaces and Regions
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
A ϕ-FEM approach for time-dependent domains with unfitted meshes
Université de Strasbourg, CNRS, Inria（斯特拉斯堡大学，法国国家科学研究中心，法国信息与自动化研究所）；Université Marie et Louis Pasteur, CNRS（玛丽·路易·巴斯德大学，法国国家科学研究中心）；University of Göttingen（哥廷根大学）
A High-Order Surface Finite Element Method Based on Intersections with Background Tetrahedral Meshes
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
Geometry-Preserving Reduced-Order Modeling via Immersed Tensor Decomposition (ITD)
University of Chinese Academy of Sciences（中国科学院大学）；Northwestern University（西北大学）；HIDENN-AI, Inc（HIDENN-AI公司）；Oden Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）
A stabilized trace finite element method for partial differential equations on evolving surfaces
University of Göttingen（哥廷根大学）；University of Houston（休斯顿大学）；Sechenov University（谢切诺夫大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所，国家数学与交叉科学中心）
