The virtual element method on polygonal pixel-based tessellations

TL;DR

This paper demonstrates that the virtual element method, combined with boundary correction, effectively solves problems on polygonal pixel-based tessellations, enabling higher order accuracy and efficiency for domains from images.

Contribution

It introduces a boundary correction technique and a static condensation strategy that together enhance the stability and efficiency of the virtual element method on pixel-based tessellations.

Findings

Polygonal elements satisfy stability assumptions for any polynomial order.

The method achieves higher order accuracy on pixel-based tessellations.

The static condensation strategy improves computational efficiency.

Abstract

We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains. We focus on the case of approximating domains obtained as the union of squared elements out of a uniform structured mesh, such as the one that naturally arises when the domain is issued from an image. We show, both theoretically and numerically, that resorting to polygonal elements allows the assumptions required for stability to be satisfied for any polynomial order. This allows us to fully exploit the potential of higher order methods. Efficiency is ensured by a novel static condensation strategy acting on the edges of the decomposition.