Isoparametric Virtual Element Methods

TL;DR

This paper introduces two novel isoparametric Virtual Element Methods for solving linear elliptic PDEs on complex 2D domains, demonstrating optimal convergence and practical effectiveness.

Contribution

It proposes two new approaches for isoparametric Virtual Element Methods of arbitrary order, enhancing accuracy on curved domains with proven optimal convergence.

Findings

Both methods converge optimally.

Methods are effective on curved domains.

Practical implementation confirms theoretical results.

Abstract

We present two approaches to constructing isoparametric Virtual Element Methods of arbitrary order for linear elliptic partial differential equations on general two-dimensional domains. The first method approximates the variational problem transformed onto a computational reference domain. The second method computes a virtual domain and uses bespoke polynomial approximation operators to construct a computable method. Both methods are shown to converge optimally, a behaviour confirmed in practice for the solution of problems posed on curved domains.