Stabilized Finite Element Method for 3D Navier-Stokes Equations with Physical Boundary Conditions

Transcription

1 Stabilized Finite Element Method for 3D Navier-Stokes Equations with Physical Boundary Conditions Mohamed Amara, Daniela Capatina, David Trujillo Université de Pau et des Pays de l Adour Laboratoire de Mathématiques Appliquées-CNRS UMR5142 Journée GdR MOMAS - Paris - 4 décembre 2008 Stabilized Finite Element Method for 3D Navier-Stokes Equations with Physical Boundary Conditions p. 1/4

2 Mathematical Framework 2D : stationary incompressible Navier-Stokes eqs. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 2/4

3 Mathematical Framework 2D : stationary incompressible Navier-Stokes eqs. Boundary conditions: u n = 0, u t = 0 on Γ 1 p u u = p 0, u t = 0 on Γ 2 u n = 0, ω = ω 0 on Γ 3 with ω = curlu the scalar vorticity (see also Conca et al. IJNMF 95, Dubois M3AS 02) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 2/4

4 Mathematical Framework 2D : stationary incompressible Navier-Stokes eqs. Boundary conditions: u n = 0, u t = 0 on Γ 1 p u u = p 0, u t = 0 on Γ 2 u n = 0, ω = ω 0 on Γ 3 with ω = curlu the scalar vorticity (see also Conca et al. IJNMF 95, Dubois M3AS 02) Amara, Capatina and Trujillo Math. Comp. 07 : Three-fields formulation in (u,ω,p) thanks to : u. u = ωu + 1 (u u) 2 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 2/4

5 Mathematical Framework From now on : Ω R 3 connected bounded polyhedron. Stationary incompressible Navier-Stokes equations { ν u + u. u + p = f in Ω, divu = 0 in Ω. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 3/4

6 Mathematical Framework From now on : Ω R 3 connected bounded polyhedron. Stationary incompressible Navier-Stokes equations { ν u + u. u + p = f in Ω, divu = 0 in Ω. Boundary conditions: u n = 0, u n = 0 on Γ 1 p u u = p 0, u n = 0 on Γ 2 u n = 0, ω n = ω 0 on Γ 3 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 3/4

7 Mathematical Framework From now on : Ω R 3 connected bounded polyhedron. Stationary incompressible Navier-Stokes equations { ν u + u. u + p = f in Ω, divu = 0 in Ω. Boundary conditions: u n = 0, u n = 0 on Γ 1 p u u = p 0, u n = 0 on Γ 2 u n = 0, ω n = ω 0 on Γ 3 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 3/4

8 Mathematical Framework We take: f L 4 3(Ω), ω 0 = 0, p 0 = 0 and Γ 2 > 0. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 4/4

9 Mathematical Framework We take: f L 4 3(Ω), ω 0 = 0, p 0 = 0 and Γ 2 > 0. Dynamic pressure: p = p u u Vorticity: ω = curlu Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 4/4

10 Mathematical Framework We take: f L 4 3(Ω), ω 0 = 0, p 0 = 0 and Γ 2 > 0. Dynamic pressure: p = p u u Vorticity: ω = curlu By means of the relation : u u + p = p + ω u, the system becomes: νcurlω + p + ω u = f in Ω, ω = curlu in Ω, divu = 0 in Ω. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 4/4

11 Mathematical Framework We take: f L 4 3(Ω), ω 0 = 0, p 0 = 0 and Γ 2 > 0. Dynamic pressure: p = p u u Vorticity: ω = curlu By means of the relation : u u + p = p + ω u, the system becomes: Unknowns: vector fields: scalar field: p. νcurlω + p + ω u = f in Ω, ω = curlu in Ω, divu = 0 in Ω. u, ω Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 4/4

12 The linear Stokes operator Associated Stokes problem : νcurlω + p = g in Ω, ω = curlu in Ω, divu = 0 in Ω, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 5/4

13 The linear Stokes operator Associated Stokes problem : Boundary conditions νcurlω + p = g in Ω, ω = curlu in Ω, divu = 0 in Ω, u n = 0, u n = 0 on Γ 1, p = 0, u n = 0 on Γ 2, u n = 0, ω n = 0 on Γ 3. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 5/4

14 The linear Stokes operator Associated Stokes problem : Boundary conditions Hypothesis: g L 4 3(Ω). νcurlω + p = g in Ω, ω = curlu in Ω, divu = 0 in Ω, u n = 0, u n = 0 on Γ 1, p = 0, u n = 0 on Γ 2, u n = 0, ω n = 0 on Γ 3. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 5/4

15 The linear Stokes operator Mixed variational formulation: Find (σ,u) X M suchthat a(σ,τ) + b(τ,u) = 0 τ X, b(σ,v) = l(v) v M, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 6/4

16 The linear Stokes operator Mixed variational formulation: Find (σ,u) X M suchthat a(σ,τ) + b(τ,u) = 0 τ X, b(σ,v) = l(v) v M, for all σ = (ω,p), τ = (θ,q) X and v M : a(σ,τ) = ν ω.θdω, Ω b(τ, v) = ν θ.curlvdω + Ω l(v) = g.vdω. Ω Ω qdivvdω, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 6/4

17 The linear Stokes operator The following Hilbert spaces are employed : M = {v H(div,curl; Ω); v n Γ1 Γ 3 = 0,v n Γ1 Γ 2 = 0}, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 7/4

18 The linear Stokes operator The following Hilbert spaces are employed : M = {v H(div,curl; Ω); v n Γ1 Γ 3 = 0,v n Γ1 Γ 2 = 0}, X = L 2 (Ω) L 2 (Ω), Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 7/4

19 The linear Stokes operator The following Hilbert spaces are employed : M = {v H(div,curl; Ω); v n Γ1 Γ 3 = 0,v n Γ1 Γ 2 = 0}, X = L 2 (Ω) L 2 (Ω), where H(div,curl; Ω) = {v L 2 (Ω);divv L 2 (Ω),curlv L 2 (Ω)}. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 7/4

20 The linear Stokes operator The following Hilbert spaces are employed : M = {v H(div,curl; Ω); v n Γ1 Γ 3 = 0,v n Γ1 Γ 2 = 0}, X = L 2 (Ω) L 2 (Ω), where H(div,curl; Ω) = {v L 2 (Ω);divv L 2 (Ω),curlv L 2 (Ω)}. H(div,curl; Ω) and M are both normed by v M = ( v 2 0,Ω + divv 2 0,Ω + curlv 2 0,Ω )1/2. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 7/4

21 The linear Stokes operator We introduce v M = ( divv 2 0,Ω + curlv 2 0,Ω )1/2. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 8/4

22 The linear Stokes operator We introduce v M = ( divv 2 0,Ω + curlv 2 0,Ω )1/2. We assume that: M is equivalent to the norm M in M, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 8/4

23 The linear Stokes operator We introduce v M = ( divv 2 0,Ω + curlv 2 0,Ω )1/2. We assume that: M is equivalent to the norm M in M, M is compactly imbedded in L p (Ω), p > 4 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 8/4

24 The linear Stokes operator We introduce v M = ( divv 2 0,Ω + curlv 2 0,Ω )1/2. We assume that: M is equivalent to the norm M in M, M is compactly imbedded in L p (Ω), p > 4 the traces of the elements of M belong to L 2 (Γ). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 8/4

25 The linear Stokes operator We introduce v M = ( divv 2 0,Ω + curlv 2 0,Ω )1/2. We assume that: M is equivalent to the norm M in M, M is compactly imbedded in L p (Ω), p > 4 the traces of the elements of M belong to L 2 (Γ). Last two assumptions hold if : M H s (Ω), 3 4 < s 1 (in 2D, we can prove : M H s (Ω) with 1 2 < s 1) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 8/4

26 The linear Stokes operator Continuous linear Stokes operator S defined by: S : L 4/3 (Ω) g X L 4 (Ω) S(g) = (σ,u). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 9/4

27 The linear Stokes operator Babuška-Brezzi: find admits a unique solution if: inf sup v M\{0} σ X (σ,u) X M suchthat a(σ,τ) + b(τ,u) = 0 τ X, b(σ,v) = l(v) v M, b(σ, v) v M σ X γ > 0 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 10/4

28 The linear Stokes operator Babuška-Brezzi: find admits a unique solution if: inf sup v M\{0} σ X (σ,u) X M suchthat a(σ,τ) + b(τ,u) = 0 τ X, b(σ,v) = l(v) v M, b(σ, v) v M σ X γ > 0 a(.,.) is V elliptic, where V = {τ X; b(τ,v) = 0, v M} Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 10/4

29 The nonlinear Navier-Stokes operator We introduce the nonlinear operator G : X L 4 (Ω) L 4/3 (Ω) G(τ,v) = θ v, where τ = (θ,q). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 11/4

30 The nonlinear Navier-Stokes operator We introduce the nonlinear operator G : X L 4 (Ω) L 4/3 (Ω) G(τ,v) = θ v, where τ = (θ,q). Navier-Stokes equations: where F is defined by : F(σ,u) = 0, F : X L 4 (Ω) X L 4 (Ω) F(τ,v) = (τ,v) S(f G(τ,v)). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 11/4

31 The nonlinear Navier-Stokes operator We assume that there exists a solution (σ,u) such that: F(σ,u) = 0 and DF(σ,u) is an isomorphism on X L 4 (Ω), Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 12/4

32 The nonlinear Navier-Stokes operator We assume that there exists a solution (σ,u) such that: F(σ,u) = 0 and DF(σ,u) is an isomorphism on X L 4 (Ω), where and DF(σ,u) = Id + S(DG(σ,u)). DG(σ,u)(τ,v) = θ u+ ω v τ = (θ,q) X. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 12/4

33 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

34 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, E h the set of internal faces, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

35 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, E h the set of internal faces, h K the diameter of the tetrahedron K, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

36 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, E h the set of internal faces, h K the diameter of the tetrahedron K, h e the diameter of the face e, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

37 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, E h the set of internal faces, h K the diameter of the tetrahedron K, h e the diameter of the face e, M h = {v h M; K T h, v h K P 1 (K)} C 0 (Ω), Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

38 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, E h the set of internal faces, h K the diameter of the tetrahedron K, h e the diameter of the face e, M h = {v h M; K T h, v h K P 1 (K)} C 0 (Ω), L h = { q h L 2 (Ω); K T h, q h K P 0 (K) }. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

39 The discrete Stokes operator We define (T h ) h>0 regular family of triangulations of Ω consisting of tetrahedrons, E h the set of internal faces, h K the diameter of the tetrahedron K, h e the diameter of the face e, M h = {v h M; K T h, v h K P 1 (K)} C 0 (Ω), L h = { q h L 2 (Ω); K T h, q h K P 0 (K) }. X h = L h L h and M h are discrete subspaces of X and M. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 13/4

40 The discrete Stokes operator S h S h : L 4 3 (Ω) Xh M h g (σ h,u h ) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 14/4

41 The discrete Stokes operator S h S h : L 4 3 (Ω) Xh M h g (σ h,u h ) (σ h,u h ) solution of : Find (σ h = (ω h,p h ),u h ) X h M h suchthat a(σ h,τ h ) + βa h (σ h,τ h ) + b(τ h,u h ) = 0 τ h = (θ h,q h ) X h, b(σ h,v h ) = Ω g.v hdx v h M h, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 14/4

42 The discrete Stokes operator S h (σ h,u h ) solution of : S h : L 4 3 (Ω) Xh M h g (σ h,u h ) Find (σ h = (ω h,p h ),u h ) X h M h suchthat a(σ h,τ h ) + βa h (σ h,τ h ) + b(τ h,u h ) = 0 τ h = (θ h,q h ) X h, b(σ h,v h ) = Ω g.v hdx v h M h, where A h (σ h,τ h ) = e E h h e e [p h ][q h ]ds + e Γ 2 h e e p h q h dγ, β > 0 stabilization parameter, [.] the jump across the edge e E h. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 14/4

43 The discrete Stokes operator The inf-sup condition is satisfied. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 15/4

44 The discrete Stokes operator The inf-sup condition is satisfied. Coercivity: we add A h to the bilinear form a(, ). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 15/4

45 The discrete Stokes operator The inf-sup condition is satisfied. Coercivity: we add A h to the bilinear form a(, ). a + βa h is uniformly continuous on X h. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 15/4

46 The discrete Stokes operator The inf-sup condition is satisfied. Coercivity: we add A h to the bilinear form a(, ). a + βa h is uniformly continuous on X h. a + βa h is uniformly elliptic on V h. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 15/4

47 The discrete Stokes operator The inf-sup condition is satisfied. Coercivity: we add A h to the bilinear form a(, ). a + βa h is uniformly continuous on X h. a + βa h is uniformly elliptic on V h. Existence and uniqueness for discrete Stokes pb. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 15/4

48 The discrete Stokes operator The operator S h is linear, continuous and satisfies: S h (g) X L 4 (Ω) c g L 4/3 (Ω) with c independent of h and Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 16/4

49 The discrete Stokes operator The operator S h is linear, continuous and satisfies: S h (g) X L 4 (Ω) c g L 4/3 (Ω) with c independent of h and g L 4/3 (Ω), lim (S S h )(g) X L 4 (Ω) = 0. h 0 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 16/4

50 The discrete Stokes operator The operator S h is linear, continuous and satisfies: S h (g) X L 4 (Ω) c g L 4/3 (Ω) with c independent of h and g L 4/3 (Ω), lim (S S h )(g) X L 4 (Ω) = 0. h 0 Moreover, if g L 2 (Ω) and (σ,u) H 1 (Ω) H 2 (Ω) : (S S h )(g) X M ch g L2 (Ω). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 16/4

51 The discrete Stokes operator Proof of the inf-sup condition: (usually difficult) Since, for all τ h = (θ h,q h ) X h taking b(τ h,v h ) = (θ h,curlv h ) + (q h,div v h ), we obtain τ h = ( curlv h,divv h ) X h b(τ h,v h ) = τ h 2 X = curlv h 2 0,Ω + divv h 2 0,Ω = v h 2 M and sup τ h X h b(τ h,v h ) τ h X = b(τ h,v h ) τ h X = τ h x = v h M. Inf-Sup condition verified with constant γ = 1. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 17/4

52 The discrete Stokes operator Proof of the V h - coercivity (usually trivial) of a h : a h = a + βa h. Semi-norm on X h associated to A h : for all τ h X h, τ h h = A h (τ h,τ h ) = ( e C h h e [q h ] 2 0, e )1 2 We have for all τ h = (θ h,q h ) V h : a h (τ h,τ h ) = θ h 2 0,Ω + β τ h 2 h α τ h 2 x Proof of the X h - continuity of a h : for all τ h X h, τ h h c q h 0,Ω. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 18/4

53 The discrete Navier-Stokes problem The discrete Navier-Stokes formulation can be written: where F h is defined by : F h (σ h,u h ) = (0,0) F h : X L 4 (Ω) X L 4 (Ω) F h (τ,v) = (τ,v) S h (f G(τ,v)). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 19/4

54 The discrete Navier-Stokes problem The discrete Navier-Stokes formulation can be written: where F h is defined by : F h (σ h,u h ) = (0,0) F h : X L 4 (Ω) X L 4 (Ω) F h (τ,v) = (τ,v) S h (f G(τ,v)). The functional F h is differentiable and: DF h (σ h,u h ) = Id + S h (DG(σ h,u h )). Main tool: variant of the implicit function theorem. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 19/4

55 The discrete Navier-Stokes problem We show that: (τ,v) Y = X L 4 (Ω), DF h (τ,v) DF h (τ,v) c (τ,v) (τ,v) Y. lim h 0 F h (σ,u) Y = 0. h 0, h h 0,DF h (σ,u) is an isomorphism and DFh (σ,u) 1 2 DF(σ,u) 1. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 20/4

56 The discrete Navier-Stokes problem Then Uniqueness for h < h 0 in a neighborhood of (σ,u). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 21/4

57 The discrete Navier-Stokes problem Then Uniqueness for h < h 0 in a neighborhood of (σ,u). a priori estimates: (σ,u) (σ h,u h ) Y c F h (σ,u) Y. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 21/4

58 The discrete Navier-Stokes problem Then Uniqueness for h < h 0 in a neighborhood of (σ,u). a priori estimates: (σ,u) (σ h,u h ) Y c F h (σ,u) Y. a posteriori estimates (σ,u) (σ h,u h ) Y c F(σ h,u h ) Y. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 21/4

59 A priori estimates We get : Unconditionally convergent method, since F h (σ,u) = (S h S)(f ω u). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 22/4

60 A priori estimates We get : Unconditionally convergent method, since F h (σ,u) = (S h S)(f ω u). Optimal convergence rate O(h) if smooth solution Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 22/4

61 A priori estimates We get : Unconditionally convergent method, since F h (σ,u) = (S h S)(f ω u). Optimal convergence rate O(h) if smooth solution Aubin-Nitsche argument u u h L4 (Ω) O(h5/4 ) (respt. O(h 3/2 ) in 2D) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 22/4

62 A posteriori estimators Residuals on every element K of the triangulation: Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 23/4

63 A posteriori estimators Residuals on every element K of the triangulation: η 1 = ν(ω h curlu h ), η 2 = divu h,η 3 = f ω h u h, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 23/4

64 A posteriori estimators Residuals on every element K of the triangulation: η 4 = η 1 = ν(ω h curlu h ), η 2 = divu h,η 3 = f ω h u h, ν[ω h ] if e E h νω h if e Γ 3,η 5 = 0 if e Γ 1 Γ 2 [p h ] if e E h p h if e Γ 2, 0 if e Γ 1 Γ 3 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 23/4

65 A posteriori estimators Residuals on every element K of the triangulation: η 4 = η 1 = ν(ω h curlu h ), η 2 = divu h,η 3 = f ω h u h, ν[ω h ] if e E h νω h if e Γ 3,η 5 = 0 if e Γ 1 Γ 2 ηk 2 = η 1 2 0,K + η 2 2 0,K +h2s K η 3 2 0,K +h2s 1 e [p h ] if e E h p h if e Γ 2, 0 if e Γ 1 Γ 3 e K where s ] 3 4, 1 ] is such that M H s (Ω). Then : ( η 4 2 0,e + η 5 2 0,e ) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 23/4

66 A posteriori estimators Residuals on every element K of the triangulation: η 4 = η 1 = ν(ω h curlu h ), η 2 = divu h,η 3 = f ω h u h, ν[ω h ] if e E h νω h if e Γ 3,η 5 = 0 if e Γ 1 Γ 2 ηk 2 = η 1 2 0,K + η 2 2 0,K +h2s K η 3 2 0,K +h2s 1 e [p h ] if e E h p h if e Γ 2, 0 if e Γ 1 Γ 3 e K where s ] 3 4, 1 ] is such that M H s (Ω). Then : (σ,u) (σ h,u h ) Y c( K T h η 2 K )1/2. ( η 4 2 0,e + η 5 2 0,e ) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 23/4

67 A posteriori estimators: A 2D example Coarse grid Refined grid Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 24/4

68 A posteriori estimators: A 2D example Coarse grid Refined grid Estimators on the different meshes Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 24/4

69 A tool to fix the parameter β A posteriori estimators Example on the 2D step test Values of η Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 25/4

70 A tool to fix the parameter β A posteriori estimators Example on the 2D step test Values of η for β = 0.03, Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 25/4

71 A tool to fix the parameter β A posteriori estimators Example on the 2D step test Values of η for β = 0.03, β = 0.5 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 25/4

72 A tool to fix the parameter β A posteriori estimators Example on the 2D step test Values of η for β = 0.03, β = 0.5 and β = 2 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 25/4

73 Numerical results: Convergence Rate Ω =] 1, 1[ 3, p = sin(πx)sin(πy)sin(πz), u 1 = cos(πx)sin(πy)sin(πz), u 2 = sin(πx)cos(πy)sin(πz), u 3 = 2sin(πx)sin(πy)cos(πz). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 26/4

74 Numerical results: Convergence Rate Ω =] 1, 1[ 3, p = sin(πx)sin(πy)sin(πz), u 1 = cos(πx)sin(πy)sin(πz), u 2 = sin(πx)cos(πy)sin(πz), u 3 = 2sin(πx)sin(πy)cos(πz). Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 26/4

75 Numerical results: Convergence Rate Ω =] 1, 1[ 3, p = sin(πx)sin(πy)sin(πz), u 1 = cos(πx)sin(πy)sin(πz), u 2 = sin(πx)cos(πy)sin(πz), u 3 = 2sin(πx)sin(πy)cos(πz). Slope : 1 3 and h N 1 3 then errors O(h) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 26/4

76 Numerical results: The cavity tests (Re=100) The cavity test on the unit cube Mesh: 3232 elem., 838 nodes Streamlines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 27/4

77 Numerical results: The cavity tests (Re=100) Vorticity lines Pressure isolines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 28/4

78 Numerical results: The cavity tests (Re=5000) The cavity test on the domain ]0,1[ ]0,1[ ]0,2[ Mesh: 8619 elem., 1920 nodes Vorticity lines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 29/4

79 Numerical results: The cavity tests (Re=5000) Velocity Streamlines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 30/4

80 Numerical results: The step test (Re=10) Mesh :10794 elements, 2665 nodes Pressure imposed on the inlet and outlet boundaries. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 31/4

81 Numerical results: The step test (Re=10) Velocity Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 32/4

82 Numerical results: The step test (Re=10) Streamlines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 33/4

83 Numerical results: The step test (Re=10) Pressure Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 34/4

84 Numerical results: The step test (Re=1000) Mesh :10794 elements, 2665 nodes Pressure imposed on the inlet and outlet boundaries. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 35/4

85 Numerical results: The step test (Re=1000) Streamlines closed to the step Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 36/4

86 Numerical results: The step test (Re=1000) View from the bottom Lateral view Velocity near the step Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 37/4

87 Numerical results: The step test (Re=1000) Streamlines closed to the step Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 38/4

88 Numerical results: The step test (Re=1000) Streamlines closed to the step Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 39/4

89 Numerical results: The step test (Re=1000) Vorticity lines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 40/4

90 Numerical results: The step test (Re=1000) Pressure Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 41/4

91 Numerical results: T-shaped domain (Re=100) Mesh: elem., 2469 nodes Pressure (imposed on inlet and outlet) Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 42/4

92 Numerical results: T-shaped domain (Re=100) Streamlines Velocity Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 43/4

93 Numerical results: T-shaped domain (Re=100) Vorticity lines Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 44/4

94 Numerical results: T-shaped domain (Re=10e4) Mesh: elem., 5024 nodes Pressure imposed on the inlet and outlet boundaries. Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 45/4

95 Numerical results: T-shaped domain (Re=10e4) Streamlines and Velocity Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 46/4

96 Numerical results: T-shaped domain (Re=10e4) Vorticity lines and pressure Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 47/4

97 Numerical results: T-shaped domain Streamlines for Re=100 and Re=10000 Stabilized Finite Element Method for 3D Navier-Stokes Equations p. 48/4