Dual mixed finite element method of the elasticity and elastodynamic problems: a priori and a posteriori error analysis.
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(2) Universit´ e de Valenciennes et du Hainaut Cambr´ esis ´ D´ epartement de Math´ ematiques Ecole doctorale LAMIH Valenciennes. M´ ethode des ´ el´ ements finis mixte duale pour les probl` emes de l’´ elasticit´ e et de l’´ elastodynamique: analyse d’erreur ` a priori et ` a posteriori ` THESE pr´esent´ee et soutenue publiquement le 10 juillet 2006 pour l’obtention du. Doctorat de l’universit´ e de Valenciennes et de Hainaut Cambr´ esis (sp´ ecialit´ e: Math´ ematiques Appliqu´ ees) par. Lahcen BOULAAJINE. Composition du jury Pr´esident :. Serge Nicaise. Universit´e de Valenciennes. Directeur de Th`ese :. Luc Paquet. Universit´e de Valenciennes. Rapporteurs :. Christine Bernardi Barbara Wolmuth. Universit´e Pierre-et-Marie-Curie Universit´e de Stuttgart. Examinateurs :. Caterina Calgaro F´elix Ali-Mehmeti Serge Nicaise. Universit´e de Lille Universit´e de Valenciennes Universit´e de Valenciennes. Invit´e :. Philippe Bouillard. Universit´e Libre de Bruxelles. Laboratoire de Math´ ematiques et ses appliquations de Valenciennes — EA 4015.
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(5) Universit´ e de Valenciennes et du Hainaut Cambr´ esis ´ D´ epartement de Math´ ematiques Ecole doctorale LAMIH Valenciennes. M´ ethode des ´ el´ ements finis mixte duale pour les probl` emes de l’´ elasticit´ e et de l’´ elastodynamique: analyse d’erreur ` a priori et ` a posteriori ` THESE pr´esent´ee et soutenue publiquement le 10 juillet 2006 pour l’obtention du. Doctorat de l’universit´ e de Valenciennes et de Hainaut Cambr´ esis (sp´ ecialit´ e: Math´ ematiques Appliqu´ ees) par. Lahcen BOULAAJINE. Composition du jury Pr´esident :. Serge Nicaise. Universit´e de Valenciennes. Directeur de Th`ese :. Luc Paquet. Universit´e de Valenciennes. Rapporteurs :. Christine Bernardi Barbara Wolmuth. Universit´e Pierre-et-Marie-Curie Universit´e de Stuttgart. Examinateurs :. Caterina Calgaro F´elix Ali-Mehmeti Serge Nicaise. Universit´e de Lille Universit´e de Valenciennes Universit´e de Valenciennes. Invit´e :. Philippe Bouillard. Universit´e Libre de Bruxelles. Laboratoire de Math´ ematiques et ses appliquations de Valenciennes — EA 4015.
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(319) Ω, − σ (u) = f s S "X "S u u = 0 Γ , . j. e. j. N. e. D. D. N. D. j. D. N. j. 1. 1. 1. 2. 2. 2. X. s. . 2. σs (u).n. = 0. . D. ΓN .. 2.
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(332)
(333). . (u). ! . ω :=. Σ0 := (τ, q) ∈ [L2 (Ω)]2×2 × [L2 (Ω)];. 1 2. . 1 ∂u2 ∂u1 (u) := ( − ) 2 ∂x1 ∂x2. . s S " X "^U u. (τ − qδ) ∈ [L2 (Ω)]2 ,. (τ − qδ).n = 0 ΓN . V × W := ( , θ) ∈ [L2 (Ω)]2 × L2 (Ω) ,. . . s S " X "ZX u. W ! b ! L : 48 X X e" GX3Y e" GXZ
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(345) 0,h. h. h. 0. h. 1. 2×2. 1. 1. 2. 2. 1. . (τ ). :=. (τ ). :=. (τ ). :=. (τ ). :=. . (τ ). :=. . (v). :=.
(346) . (φ). :=. . . . . (v). :=. τ11 + τ22 , T ∂τ11 ∂τ12 ∂τ21 ∂τ22 + , + , ∂x1 ∂x2 ∂x1 ∂x2 τ21 − τ12 , 1 τ + τT , 2 T ∂τ12 ∂τ11 ∂τ22 ∂τ21 , − , − ∂x1 ∂x2 ∂x1 ∂x2 ∂v1 ∂v1 − ∂x2 ∂x1 ∂v ∂v2 , 2 − ∂x2 ∂x1 T ∂φ ∂φ , − ∂x2 ∂x1 ∂v2 ∂v1 − . ∂x1 ∂x2. * 8 ! # :% #S2 W !
(347) 8 s S " X "PZqu$W + L : 8 0# s S " X " ` u " b# ! # +8 /W # 8 8 ! 8 $0 vs . X S u
(348). !J JN#5]d45 1. [H (Ω)]. . 6
(349) . 2. . . . τ ∈ [H(. . , Ω)]2 :=. ((v), τ ) =< τ n, v > −( τ n = (τ11 n1 + τ12 n2 , τ21 n1 + τ22 n2 ). . . . τ ∈ [L2 (Ω)]2×2 ;. . . 1 . τ, v) − ( (τ ), 2. τ ∈ [L2 (Ω)]2. v),. . . v ∈. # ! !W 8 W ! ! (σ, p) " (τ, q) ∈ Σ 2^= σ = (σ, p) " τ = (τ, q) ! 5 M=
(350) (u, ω) " (v, θ) ∈ V × W 2^= u = (u, ω) " v = (v, θ) " 3 ! W 2 D8 a : Σ ×Σ → T .b" b : Σ ×(V ×W ) → T . ! D8 F : V ×W → T . h
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