Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

TL;DR

This paper introduces a geometrically intrinsic arbitrary-order Virtual Element Method for solving elliptic PDEs on surfaces, avoiding explicit surface geometry approximation and extending classical VEM properties to anisotropic discretizations.

Contribution

It develops a novel intrinsic VEM framework for surface PDEs that handles polygonal meshes without explicit surface approximation.

Findings

Method performs well on triangular and polygonal meshes

Theoretical properties are validated through extensive tests

Limitations depend on surface regularity and approximation quality

Abstract

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.