Schémas à deux-grilles pour la résolution du problème de Navier-Stokes instationnaire incompressible
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(2) Ecole doctorale des Sciences Math´ ematiques de Paris Centre UFR 921. Universit´ e Saint–Joseph Facult´ e des Sciences. Sch´ emas ` a deux grilles pour la r´ esolution du probl` eme de Navier-Stokes instationnaire incompressible ` THESE pr´esent´ee et soutenue publiquement le 03 juillet 2006 pour l’obtention du titre de. Docteur de l’Universit´ e Pierre et Marie Curie – Paris VI et Docteur de l’Universit´ e Saint-Joseph – Beyrouth (sp´ ecialit´ e Math´ ematiques Appliqu´ ees) par. Hyam ABBOUD Composition du jury Rapporteurs :. Jean-Luc GUERMOND Rapha¨ele HERBIN. Examinateurs :. Vivette GIRAULT (Directrice de th`ese) Fr´ederic HECHT Charbel KLAIANY Nabil NASSIF Toni SAYAH (Directeur de th`ese). Laboratoire Jacques-Louis Lions UMR 7598. D´ epartement de Math´ ematiques Facult´ e des Sciences.
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(441) η ∆t, ) :. N n=0. 1/2 2 ∆t vηn − v(tn )L2 (Ω) ≤ C(η 3 + (∆t)2 + η(∆t)2 ).. 7JV8 KJ.
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(445) . N n=0. 1/2 2 ∆t vηn − v(tn )L2 (Ω) ≤ Cη 3 .. 7JS8 7JL8.
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(454) 7JS8 2 2 6%, < L (Ω×]0, T [) B −1 N n=0. 1/2 2 ∆t un+1 − u(tn+1 )L2 (Ω) ≤ Cη 3 . η. 7TD8. -
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(462) :. n−1 sup unh − u(tn )L2 (Ω) + sup 2(unh − u(tn )) − (un−1 − u(t )) h L2 (Ω). 1≤n≤N. +. 1≤n≤N. N 2 n δ (u − u(tn ))2 2 h. 1/2. L (Ω). +. N 1/2 √ ν ∆t|unh − u(tn )|2H 1 (Ω). n=1. 7TK8. n=1. ≤ C(H 3 + h2 + (∆t)2 + H(∆t)2 ),. . N n=1. KT. ∆t p(tn ) − pnh 2L2 (Ω). 1/2. ≤ C(h2 + H 3 + (∆t)2 ),. 7TC8.
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(465)
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(468) h, H ∆t. , α γ, h H . B. 7TJ8 C(H 3 + h2 + (∆t)2 + H(∆t)2) ≤ Ch2 7 7TK88 C(h2 + H 3 + (∆t)2) ≤2 Ch2 7 7TC88 B . & h α H 3 ≤ h2 ≤ γ H 3 ,. ! % .
(469)
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(481) (u, p) = (rot ψ, p) \ ψ p . B 2 (x+y). ψ(t, x, y) = te−t. y2 (1 − y)2 sin2 (πx),. p(t, x, y) = te−t cos(2πx) sin(2πy).. # & -
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