AIMS Mathematics, 2018, 3(4): 485-513. doi: 10.3934/Math.2018.4.485

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Stabilized bi-grid projection methods in finite elements for the 2D incompressible Navier-Stokes equations

1 Département de math´ematiques, Facult´e des Sciences II, Universit´e Libanaise, Fanar, Liban
2 Laboratoire Ami´enois de Math´ematiques Fondamentales et Appliqu´ees (LAMFA), UMR CNRS7352, Universit´e de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens France
3 Laboratoire de Physique Appliqu´ee (LPA), Facult´e des Sciences II, Universit´e Libanaise, Fanar,Liban

We introduce a family of bi-grid schemes in finite elements for solving 2D incompressible Navier-Stokes equations in velocity and pressure (u; p). The new schemes are based on projection methods and use two pairs of FEM spaces, a sparse and a fine one. The main computational e ortis done on the coarsest velocity space with an implicit and unconditionally time scheme while itscorrection on the finer velocity space is realized with a simple stabilized semi-implicit scheme whosethe lack of stability is compensated by a high mode stabilization procedure; the pressure is updatedusing the free divergence property. The new schemes are tested on the lid driven cavity up to Re =7500. An enhanced stability is observed as respect to classical semi-implicit methods and an importantgain of CPU time is obtained as compared to implicit projection schemes.
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