A posteriori error analysis and adaptivity for a VEM discretization of the Navier-Stokes equations

TL;DR

This paper develops a residual-based a posteriori error estimator for the Virtual Element Method applied to steady Navier-Stokes equations, enabling adaptive mesh refinement and improved solution accuracy.

Contribution

It introduces a novel a posteriori error estimator for VEM discretizations of Navier-Stokes, including nonlinear term considerations, and demonstrates its effectiveness through numerical tests.

Findings

Estimator reliably bounds combined velocity and pressure errors.

Efficiency bounds confirm the estimator's lower bounds.

Adaptive refinement improves solution accuracy and convergence.

Abstract

We consider the Virtual Element method (VEM) introduced by Beir\~ao da Veiga, Lovadina and Vacca in 2016 for the numerical solution of the steady, incompressible Navier-Stokes equations; the method has arbitrary order $k \geq 2$ and guarantees divergence-free velocities. For such discretization, we develop a residual-based a posteriori error estimator, which is a combination of standard terms in VEM analysis (residual terms, data oscillation, and VEM stabilization), plus some other terms originated by the VEM discretization of the nonlinear convective term. We show that a linear combination of the velocity and pressure errors is upper-bounded by a multiple of the estimator (reliability). We also establish some efficiency results, involving lower bounds of the error. Some numerical tests illustrate the performance of the estimator and of its components while refining the mesh uniformly, yielding the expected decay rate. At last, we apply an adaptive mesh refinement strategy to the computation of the low-Reynolds flow around a square cylinder inside a channel.