Tamtam Proceedings - lamsin

6 Amara et al.2.2. The nonlinear Navier-Stokes operatorWe introduce the nonlinear operatorG : X × L 4 (Ω) → L 4/3 (Ω) with G(τ, v) = θ × v ∀τ = (θ, q) ∈ X, (8)then the Navier-Stokes equations (3) can be put in the general setting of a nonlinear problemas follows :F (σ, u) = 0. (9)The mapping F is defined by :F : X × L 4 (Ω) → X × L 4 (Ω), F (τ, v) = (τ, v) − S(f − G(τ, v), ω 0 , p 0 ). (10)We assume in what follows 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).It is well-known that the Navier-Stokes problem admits at least a solution (σ, u), theuniqueness holding under a hypothesis of small data. In this last case, DF (σ, u) is clearlyan isomorphism. We remark that DF (σ, u) = Id + S(DG(σ, u), 0, 0) whereDG(σ, u)(τ, v) = θ × u+ ω × u ∀τ = (θ, q) ∈ X.3. Discrete problemThe numerical approximation of the Navier-Stokes equations needs to consider onlythe discretization of the associated Stokes problem. Let (T h ) h>0 a regular family oftriangulations of Ω consisting of tetrahedrons, we denote by E h the set of internal faces.As usually, h K represents the diameter of the tetrahedron K while h e represents thediameter of the face e. We define the discrete finite element spaces:M h = {v h ∈ M; ∀K ∈ T h , v h | K ∈ P 1 (K)} ⊂ ML h = { q h ∈ L 2 (Ω); ∀K ∈ T h , q h | K ∈ P 0 (K) } , X h = L h × L h ⊂ X.3.1. The discrete Stokes operatorThe discrete linear Stokes operator S h is defined byS h : L 4 3 (Ω) × L 2 (Γ 3 ) × L 2 (Γ 2 ) → X × L 4 (Ω) with S h (g, λ, µ) = (σ h , u h ) (11)(σ h , u h ) being the solution of the discrete formulation associated with problem (4) :⎧⎨ F ind (σ h = (ω h , p h ), u h ) ∈ X h × M h such thata(σ h , τ h ) + βA h (σ h , τ h ) + b(τ h , u h ) = βc h (τ h ) ∀τ h = (θ h , q h ) ∈ X h ,⎩b(σ h , v h ) = −l(v h ) ∀v h ∈ M h ,(12)TAMTAM –Tunis– 2005