Tamtam Proceedings - lamsin

234 Amoura et al.where u = rot r ψ and ω = −∆ r ψ.Hence we can conclude that this formulation is equivalentto the one derived by Glowinski and Pironneau in [1] and [2].The reduced problem: The advantage of this decoupling is to write (1) as:⎧−∆ r ω = rot(f) on Ω⎪⎨(P )⎪⎩−∆ r ψ = ωψ| Γ = 0∂ψ∂η | Γ = 0on ΩProblem (P) is exactly a Dirichlet problem for the biharmonic operator ∆ 2 r, where thesolution is the current function ψ such that:⎧⎨ −∆ 2 rψ = rot(f) on Ω(P 0 ) ψ| Γ = 0 .⎩ ∂ψ∂n |Γ = 0Problem (P 0 ) can be written in a matrix form as:DU = F,where D and F have respectively the entries a N (h i l j , h r h s ) N and f(ρ i , ζ j )Ψ i δ j .We remark that D is a symmetric positive definite matrix, therefore the gradient conjugateAlgorithm can be used. A study of the continuous problem was done by [2], and thediscrete problem was studied in [4] and [5]. Here we would like to extend this study tothe Navier-Stokes equation..4. Numerical studyIn the same way we have applied the above analysis to a nonlinear problem and analogousresults have been found. Therefore we can write an Algorithm for the nonlinearNavier Stokes equation which is written as:⎧−ν∆ r ω + ∇ r ψ.rot r ω = rot(f) on Ω⎪⎨where ∇ r ϕ =(P )⎪⎩( 1r ∂ r(rϕ)∂ z ϕ−∆ r ψ = ωψ| Γ = 0∂ψ∂η | Γ = 0)and rot r ϕ =(∂ z ϕ− 1 r ∂ r(rϕ)on Ω,)for any regular ϕ.AlgorithmWe propose the following Algorithm- Approximate ψ and {ψ m } M m=1 by: ψ m+1 = ψ m + α(ψ m − ψ m ).TAMTAM –Tunis– 2005