Tamtam Proceedings - lamsin

204 Kharrat et al.The step (3) needs also a modified formulation while looking for a less regular pressure[1]. Besides, we notice that the approximate space of the end of step-velocity u n h isnever used in practice [7] and even in the present analysis. As a consequence, we will beconcerned by the approximation spaces of ũ n h and Φn h denoted respectively by Xn h andMh n nbuilt over the mesh Thand consisting of continuous functions which are piecewisepolynomials with degrees ≥ 1. The fully discrete scheme reads now:For each n, 1 ≤ n ≤ N, find ũ n h ∈ Xn hsolution of the variational equation :(ũ n h, v h ) + ντ n (∇ũ n h , ∇v h ) = (u n−1h, v h ) + τ n (f n , v h ) ∀v h ∈ X n h , (4)and Φ n h ∈ M h n solution of the Poisson problemand we set(∇Φ n h , ∇q h ) = − 1τ n(q h , ∇ũ n h) ∀q h ∈ M n h , (5)u n h = ũ n h − τ n ∇Φ n h , (6)initialized by u 0 h = Π hu 0 , where Π h denotes an appropriate interpolation or projectionoperator with values in { v h ∈ Xh 0; (v h, ∇q h ) = 0 ∀q h ∈ Mh} 0 . We mention that, thepair sequence of spaces (Xh n, M h n) 1≤n≤N τmust satisfy the Brezzi-Babŭska (or inf-sup)condition to eliminate all possible spurious pressure mode (see [6] for a priori analysis ofthe stability and convergence in time of the pressure).2. A posteriori error estimations2.1. The error estimatorsIn this section we present a posteriori residual error estimators for the error in timeand space induced respectively by the solutions of the algorithms (2)-(3) and (4)-(5)-(6).For each time step, we derive successively two types of estimators. The first ones ((7), (8)and (9)) beeing linked to time discretization and the second ones ((10), (11) and (12)) tospace discretization. They are defined for each n = 1, ..., N and K ∈ Thn(ζ n =ν τ n3) 1/2 ∣∣ũn h − ũ n−1 ∣h 1(7)ξ n = 1 2 ‖∇(τ nΦ n h − τ n−1 Φ n−1h)‖ 0 (8)˜S n =( ∫ tnt n−1‖f(s) − f n ‖ 2 0ds∥ ∥∥∥˜η n,K = h K fh n − ũn h − un−1 hτ nby) 1/2(9)+ ν∆ũ n h∥ + ν ∑ √hE ‖[⃗n E .∇ũ n0,K2h]‖ 0,E(10)E∈E KTAMTAM –Tunis– 2005