The Virtual Element Method with curved edges

TL;DR

This paper introduces a novel Virtual Element Method with curved edges that achieves optimal convergence rates for problems involving curved boundaries, improving upon polygonal approximations.

Contribution

The paper develops and analyzes a curved Virtual Element Method that attains optimal convergence rates for curved boundary problems, advancing the numerical analysis of VEM.

Findings

Curved VEM achieves optimal convergence rates.

Polygonal approximation leads to sub-optimal convergence.

Numerical results confirm theoretical predictions.

Abstract

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy $k \geq 2$, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.