Bricks for the mixed high-order virtual element method: projectors and differential operators

TL;DR

This paper develops the fundamental tools for high-order virtual element methods applied to PDEs, including projectors and differential operators, and demonstrates their effectiveness through numerical examples on fluid flow problems.

Contribution

It introduces a comprehensive framework for constructing virtual element approximations of PDEs, emphasizing the building blocks like projectors and operators for high-order accuracy.

Findings

Successful construction of virtual element spaces for PDEs

Numerical verification of high-order VEM accuracy

Application to Stokes, Darcy, and Navier-Stokes problems

Abstract

We present the essential instruments to deal with Virtual Element Method (VEM) for the resolution of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. Finally, we exploit such `briks' to construct virtual element approximations of Stokes, Darcy and Navier-Stokes problems and we provide a series of examples to numerically verify the theoretical behavior of high-order VEM.