High order VEM on curved domains

Abstract

We deal with the virtual element method (VEM) for solving the Poisson equation on a domain $\Omega$ with curved boundaries. Given a polygonal approximation ${\Omega }_h$ of the domain $\Omega$, the standard order $m$ VEM [6], for $m$ increasing, leads to a suboptimal convergence rate. We adapt the approach of [16] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of ${\Omega }_h$, which, to retain computability, is evaluated after applying the projector ${\Pi }^{\nabla }$ onto the space of polynomials. Numerical experiments confirm the theory.

arXiv comment17 pages

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