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.

我们在多边形单元上开发了任意阶虚拟元方法 (VEM) 的几何内在公式，用于椭圆表面偏微分方程 (PDE) 的数值解。PDE 首先使用适当的本地参考系统以协变形式编写。局部参数化的知识使我们能够考虑二维 VEM 方案，而无需任何表面几何的显式近似。考虑到最终离散化的高度各向异性特征，经典 VEM 的理论特性被扩展到我们的框架中。这些属性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测试。该方案的局限性被验证为表面规则性及其近似值的函数。