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.

翻译：有限元方法的最新进展允许实现具有曲边的网格单元。本文中，我们开发了在平面域中采用多连通网格单元（即带孔单元）所必需的工具。我们的重点在于高效评估由边界元-有限元方法（BEM-FEM）和虚拟元方法（VEM）中出现的隐式定义有限元函数的$H^1$半内积与$L^2$内积。此类函数可通过指定多项式拉普拉斯算子和连续狄利克雷迹来定义。我们证明，这些体积积分可简化为沿网格单元边界的积分，从而避免在单元内部进行任何计算。该简化的主要代价是求解一个相对较小的尼斯特伦系统以获得狄利克雷到诺伊曼映射，以及另两个尼斯特伦系统的求解以获得调和函数的“反拉普拉斯算子”，后者用于计算$L^2$内积。我们通过多个数值算例证明了高阶精度的可实现性。