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

翻译：以1美元为单位的加勒金配成方法用于多边形模贝,如VEM、BEM-FEM和Trefftz-FEM等,采用当地限定元素功能,这些功能被隐含地定义为具有多分子源和边界数据的Poisson问题的解决办法;最近,这些方法已经扩大,允许作为卷轴多边形的网状细胞使用。这种扩展对确定合适的二次形提出了新的挑战。我们描述了将这些隐含界定的功能的产品及其梯度的产品整合在一起的方法,这些功能及其梯度的产物将细胞的构件减少至其边界的构件。数字实验说明了拟议方法的实际表现。