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

Research output: Contribution to journalResearch articleContributedpeer-review



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.


Original languageEnglish
Article number30
Number of pages28
Publication statusPublished - Sept 2021

External IDs

Scopus 85111378457



  • Surface PDEs, Geometrically intrinsic operators, Virtual element method, Polygonal mesh, High-Order Methods