Abstract
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.
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Funding
This research was done in the framework of the Italian MIUR Award “Dipartimenti di Eccellenza 2018-2022” granted to the Department of Mathematical Sciences, Politecnico di Torino (CUP: E11G18000350001). CC is a member of the Italian INdAM-GNCS research group and was supported by the MIUR PRIN Project 201752HKH8-003.
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Communicated by: Paul Houston
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Canuto, C., Rosso, D. A posteriori error analysis and adaptivity for a VEM discretization of the Navier–Stokes equations. Adv Comput Math 49, 90 (2023). https://doi.org/10.1007/s10444-023-10081-9
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DOI: https://doi.org/10.1007/s10444-023-10081-9
Keywords
- A posteriori estimator
- Adaptivity
- Virtual element method
- Navier–Stokes equations
- Computational fluid dynamics