We consider the virtual element method (VEM) introduced by Beirão da Veiga et al. in 2016 for the numerical solution of the steady, incompressible Navier–Stokes equations; the method has arbitrary order \({k} \ge {2}\) 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.

我们考虑 Beirão da Veiga 等人提出的虚拟元素方法 (VEM)。2016 年稳定、不可压缩纳维-斯托克斯方程的数值解；该方法具有任意阶\({k} \ge {2}\)并保证无散速度。对于这种离散化，我们开发了一种基于残差的后验误差估计器，它是 VEM 分析中的标准项（残差项、数据振荡和 VEM 稳定性）以及源自非线性对流的 VEM 离散化的一些其他项的组合学期。我们证明了速度和压力误差的线性组合的上限是估计器的倍数（可靠性）。我们还建立了一些效率结果，涉及误差下限。一些数值测试说明了估计器及其组件的性能，同时均匀地细化网格，产生预期的衰减率。最后，我们应用自适应网格细化策略来计算通道内方柱体周围的低雷诺流。