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.
Highlights
Surface partial differential equations of elliptic and parabolic types are often used for the simulation of diverse phenomena in many fields of applications, for example in biology, atmospheric dynamics, and image processing [53, 55]
FE-based approaches are very successful in the numerical treatment of surface partial differential equations (PDEs), they share the limitation that an explicit form of the basis functions is required in the formulation of the method, and are restricted mostly to triangular/quadrilateral elements
This restriction is overcome by the Virtual Element Method (VEM) that was designed from the very beginning to work on generally shaped elements with high order of accuracy
Summary
Surface partial differential equations of elliptic and parabolic types are often used for the simulation of diverse phenomena in many fields of applications, for example in biology, atmospheric dynamics, and image processing [53, 55]. We avoid the difficulties related to high-order surface approximation by employing intrinsic geometry and following the approach described in [7] to adapt the virtual element technology to the surface PDE. Using this approach, we first rewrite the partial differential equation in covariant form in such a way that the geometric information, essentially the metric tensor, is completely encoded in the equation itself. With our approach the convergence theory extends straight-forwardly to surface problems without additional efforts
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