We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

中文翻译：

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曲面上椭圆方程的任意阶本征虚元方法

我们开发了多边形单元上的任意阶虚拟元素方法（VEM）的几何内在公式，用于椭圆表面偏微分方程（PDE）的数值解。首先使用适当的本地参考系统以协变形式编写PDE。局部参数化的知识使我们可以考虑二维VEM方案，而无需对表面几何形状进行任何显式近似。考虑到最终离散化的高度各向异性，经典VEM的理论特性扩展到了我们的框架。这些属性已使用制造的解决方案在三角形和多边形网格上进行了广泛的测试。验证了该方案的局限性是表面规则性及其近似性的函数。