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$H^1$-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEMand Trefftz-FEM employ local finite element functions that are implicitlydefined as solutions of Poisson problems having polynomial source and boundarydata. Recently, such methods have been extended to allow for mesh cells thatare curvilinear polygons. Such extensions present new challenges fordetermining suitable quadratures. We describe an approach for integratingproducts of these implicitly defined functions, as well as products of theirgradients, that reduces integrals on cells to integrals along their boundaries.Numerical experiments illustrate the practical performance of the proposedmethods.

Quadrature for Implicitly-defined Finite Element Functions on Curvilinear Polygons