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Recent advancements in finite element methods allows for the implementationof mesh cells with curved edges. In the present work, we develop the toolsnecessary to employ multiply connected mesh cells, i.e. cells with holes, inplanar domains. Our focus is efficient evaluation the $H^1$ semi-inner productand $L^2$ inner product of implicitly-defined finite element functions of thetype arising in boundary element based finite element methods (BEM-FEM) andvirtual element methods (VEM). These functions may be defined by specifying apolynomial Laplacian and a continuous Dirichlet trace. We demonstrate thatthese volumetric integrals can be reduced to integrals along the boundaries ofmesh cells, thereby avoiding the need to perform any computations in cellinteriors. The dominating cost of this reduction is solving a relatively smallNystrom system to obtain a Dirichlet-to-Neumann map, as well as the solution oftwo more Nystrom systems to obtain an ``anti-Laplacian'' of a harmonicfunction, which is used for computing the $L^2$ inner product. We demonstratethat high-order accuracy can be achieved with several numerical examples.

Evaluation of Inner Products of Implicitly-defined Finite Element Functions on Multiply Connected Planar Mesh Cells