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. 1/48
Mathematical Framework 2D : stationary incompressible Navier-Stokes eqs. Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 2/48
Mathematical Framework 2D : stationary incompressible Navier-Stokes eqs. u · n = 0 u · t = 0 on Γ 1 , p + 1 Boundary conditions: 2 u · u = p 0 u · t = 0 on Γ 2 � , u · n = 0 ω = ω 0 on Γ 3 , with ω = curl u the scalar vorticity (see also Conca et al . IJNMF ’95, Dubois M3AS ’02) Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 2/48
Mathematical Framework 2D : stationary incompressible Navier-Stokes eqs. u · n = 0 u · t = 0 on Γ 1 , p + 1 Boundary conditions: 2 u · u = p 0 u · t = 0 on Γ 2 � , u · n = 0 ω = ω 0 on Γ 3 , with ω = curl u 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 = ω u ⊥ + 1 2 ∇ ( u · u ) Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 2/48
Mathematical Framework From now on : Ω ⊂ R 3 connected bounded polyhedron. Stationary incompressible Navier-Stokes equations � − ν ∆ u + u . ∇ u + ∇ � p = f in Ω , div u = 0 in Ω . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 3/48
Mathematical Framework From now on : Ω ⊂ R 3 connected bounded polyhedron. Stationary incompressible Navier-Stokes equations � − ν ∆ u + u . ∇ u + ∇ � p = f in Ω , div u = 0 in Ω . u · n = 0 u × n = 0 on Γ 1 , p + 1 Boundary conditions: � 2 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. 3/48
Mathematical Framework From now on : Ω ⊂ R 3 connected bounded polyhedron. Stationary incompressible Navier-Stokes equations � − ν ∆ u + u . ∇ u + ∇ � p = f in Ω , div u = 0 in Ω . u · n = 0 u × n = 0 on Γ 1 , p + 1 Boundary conditions: � 2 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. 3/48
Mathematical Framework 4 We take: f ∈ L 3 (Ω) , ω 0 = 0 , p 0 = 0 and | Γ 2 | > 0 . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 4/48
Mathematical Framework 4 We take: f ∈ L 3 (Ω) , ω 0 = 0 , p 0 = 0 and | Γ 2 | > 0 . p + 1 Dynamic pressure: p = � Vorticity: ω = curl u 2 u · u Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 4/48
Mathematical Framework 4 We take: f ∈ L 3 (Ω) , ω 0 = 0 , p 0 = 0 and | Γ 2 | > 0 . p + 1 Dynamic pressure: p = � Vorticity: ω = curl u 2 u · u By means of the relation : u · ∇ u + ∇ � p = ∇ p + ω × u , the system becomes: ν curl ω + ∇ p + ω × u = f in Ω , ω = curl u in Ω , div u = 0 in Ω . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 4/48
Mathematical Framework 4 We take: f ∈ L 3 (Ω) , ω 0 = 0 , p 0 = 0 and | Γ 2 | > 0 . p + 1 Dynamic pressure: p = � Vorticity: ω = curl u 2 u · u By means of the relation : u · ∇ u + ∇ � p = ∇ p + ω × u , the system becomes: ν curl ω + ∇ p + ω × u = f in Ω , ω = curl u in Ω , div u = 0 in Ω . Unknowns: vector fields: u , ω scalar field: p. Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 4/48
The linear Stokes operator Associated Stokes problem : ν curl ω + ∇ p = g in Ω , ω = curlu in Ω , div u = 0 in Ω , Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 5/48
The linear Stokes operator Associated Stokes problem : ν curl ω + ∇ p = g in Ω , ω = curlu in Ω , div u = 0 in Ω , Boundary conditions 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. 5/48
The linear Stokes operator Associated Stokes problem : ν curl ω + ∇ p = g in Ω , ω = curlu in Ω , div u = 0 in Ω , Boundary conditions 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 . , 4 Hypothesis: g ∈ L 3 (Ω) . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 5/48
The linear Stokes operator Mixed variational formulation: ( σ, u ) ∈ X × M such that Find 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. 6/48
The linear Stokes operator Mixed variational formulation: ( σ, u ) ∈ X × M such that Find 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 ) = ω.θd Ω , ν Ω � � b ( τ, v ) = − ν θ. curlv d Ω + qdiv v d Ω , Ω Ω � l ( v ) = g . v d Ω . Ω Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 6/48
The linear Stokes operator The following Hilbert spaces are employed : M = { v ∈ H ( div, curl ; Ω); v · n | Γ 1 ∪ Γ 3 = 0 , v × n | Γ 1 ∪ Γ 2 = 0 } , Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 7/48
The linear Stokes operator The following Hilbert spaces are employed : M = { v ∈ H ( div, curl ; Ω); v · n | Γ 1 ∪ Γ 3 = 0 , v × n | Γ 1 ∪ Γ 2 = 0 } , X = L 2 (Ω) × L 2 (Ω) , Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 7/48
The linear Stokes operator The following Hilbert spaces are employed : M = { v ∈ H ( div, curl ; Ω); v · n | Γ 1 ∪ Γ 3 = 0 , v × n | Γ 1 ∪ Γ 2 = 0 } , X = L 2 (Ω) × L 2 (Ω) , where H ( div, curl ; Ω) = { v ∈ L 2 (Ω); div v ∈ L 2 (Ω) , curlv ∈ L 2 (Ω) } . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 7/48
The linear Stokes operator The following Hilbert spaces are employed : M = { v ∈ H ( div, curl ; Ω); v · n | Γ 1 ∪ Γ 3 = 0 , v × n | Γ 1 ∪ Γ 2 = 0 } , X = L 2 (Ω) × L 2 (Ω) , where H ( div, curl ; Ω) = { v ∈ L 2 (Ω); div v ∈ L 2 (Ω) , curlv ∈ L 2 (Ω) } . H ( div, curl ; Ω) and M are both normed by � v � M = ( � v � 2 0 , Ω + � div v � 2 0 , Ω + � curlv � 2 0 , Ω ) 1 / 2 . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 7/48
The linear Stokes operator We introduce | v | M = ( � div v � 2 0 , Ω + � curlv � 2 0 , Ω ) 1 / 2 . Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 8/48
The linear Stokes operator We introduce | v | M = ( � div v � 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. 8/48
The linear Stokes operator We introduce | v | M = ( � div v � 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. 8/48
The linear Stokes operator We introduce | v | M = ( � div v � 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. 8/48
The linear Stokes operator We introduce | v | M = ( � div v � 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. 8/48
The linear Stokes operator Continuous linear Stokes operator S defined by: S : L 4 / 3 (Ω) X × L 4 (Ω) → �→ S ( g ) = ( σ, u ) . g Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 9/48
The linear Stokes operator Babuška-Brezzi: ( σ, u ) ∈ X × M such that find a ( σ, τ ) + b ( τ, u ) = 0 ∀ τ ∈ X , b ( σ, v ) = − l ( v ) ∀ v ∈ M , admits a unique solution if: b ( σ, v ) v ∈ M \{ 0 } sup inf ≥ γ > 0 || v || M || σ || X σ ∈ X Stabilized Finite Element Method for 3D Navier-Stokes Equations – p. 10/48
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