An adaptive Virtual Element Method for the incompressible Navier-Stokes equations

  • Canuto, Claudio (Politecnico di Torino)
  • Rosso, Davide (Politecnico di Torino)

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We consider the Virtual Element method (VEM) introduced by Beir˜ao da Veiga, Lovadina and Vacca [1] for the numerical solution of the steady, incompressible Navier-Stokes equations; the method has arbitrary order k  2 and guarantees divergence-free velocities. For such discretization, we develop a residual-based a posteriori error estimator [2], 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 develop an adaptive mesh refinement strategy based on the quad-tree partition of geometric rectangles, which are handled as virtual elements (see also [3] for a similar approach). We apply it to the computation of the low-Reynolds flow around a square cylinder inside a channel [4], producing an accurate estimate of the length of the recirculation zone past the obstacle.