Quadrature for implicitly-defined finite element functions on curvilinear polygons

H1-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined solutions of Poisson problems having polynomial source boundary data. Recently, have been extended to allow for mesh cells curvilinear polygons. Such extensions present new challenges determining suitable quadratures. We describe an approach integrating products these functions, well their gradients, reduces integrals along boundaries. Numerical experiments illustrate the practical performance proposed methods.