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

Hyam Abboud; Clara Al Kosseifi; Jean-Paul Chehab; Hyam Abboud; Clara Al Kosseifi; Jean-Paul Chehab

doi:10.3934/Math.2018.4.485

AIMS Mathematics

2018, Volume 3, Issue 4: 485-513. doi: 10.3934/Math.2018.4.485

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Research article

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 CNRS 7352, 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

Received: 28 August 2018 Accepted: 18 October 2018 Published: 29 October 2018

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 ort is done on the coarsest velocity space with an implicit and unconditionally time scheme while its correction on the finer velocity space is realized with a simple stabilized semi-implicit scheme whose the lack of stability is compensated by a high mode stabilization procedure; the pressure is updated using 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 important gain of CPU time is obtained as compared to implicit projection schemes.
- Navier-Stokes equation,
- bi-grid method; stabilization,
- Chorin-Temam projection,
- separation of the scales
Citation: Hyam Abboud, Clara Al Kosseifi, Jean-Paul Chehab. Stabilized bi-grid projection methods in finite elements for the 2D incompressible Navier-Stokes equations[J]. AIMS Mathematics, 2018, 3(4): 485-513. doi: 10.3934/Math.2018.4.485

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Abstract

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 ort is done on the coarsest velocity space with an implicit and unconditionally time scheme while its correction on the finer velocity space is realized with a simple stabilized semi-implicit scheme whose the lack of stability is compensated by a high mode stabilization procedure; the pressure is updated using 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 important gain of CPU time is obtained as compared to implicit projection schemes.

References

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