The Virtual Element Method with curved edges
For researchers using VEM on curved domains or interfaces, this work solves a known accuracy bottleneck for higher-order approximations.
The paper addresses the sub-optimal convergence of Virtual Element Methods (VEM) on curved domains for polynomial degree k≥2, and proposes a curved VEM that achieves optimal convergence rates, demonstrated both theoretically and numerically.
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.