In this paper, we tackle the problem of constructing conforming Virtual Element spaces on polygons with curved edges. Unlike previous VEM approaches for curvilinear elements, the present construction ensures that the local VEM spaces contain all the polynomials of a given degree, thus providing the full satisfaction of the patch test. Moreover, unlike standard isoparametric FEM, this approach allows to deal with curved edges at an intermediate scale, between the small scale (treatable by homogenization) and the bigger one (where a finer mesh would make the curve flatter and flatter). The proposed method is supported by theoretical analysis and numerical tests.

中文翻译：

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多项式保留具有弯曲边缘的虚拟元素

在本文中，我们解决了在具有弯曲边缘的多边形上构建一致的虚拟元素空间的问题。与以前的曲线元素 VEM 方法不同，本结构确保局部 VEM 空间包含给定次数的所有多项式，从而完全满足补丁测试。此外，与标准等参数 FEM 不同，这种方法允许在小尺度（可通过均质化处理）和较大尺度（更精细的网格会使曲线越来越平坦）之间处理中间尺度的弯曲边缘。所提出的方法得到了理论分析和数值试验的支持。