Tamtam Proceedings - lamsin

REE for Stokes equations 205η n,K = h K√τn‖∇.u n h‖ 0,K + 1 2∑E∈E K√hEτ n‖[⃗n E .u n h]‖ 0,E (11)S n,K = h K ‖f n − f n h ‖ 0,K . (12)The quantities [⃗n E .∇ũ n h ] and [⃗n E.u n h] denote respectively the jump of the normal derivativesof ũ n h and the normal jump of un h through E in a direction ⃗n E, while fhn is theorthogonal projection of f n onto the space of polynomials with degree ≤ 1. Moreover,since our goal is mesh adaptivity, the triangulations {Th n}1≤n≤N τare not independentin practice, noting the fact that any triangulation Thnn−1can be derived from Thby locallyrefining or coarsening the mesh. On the other hand, several triangulations can beemployed at the same time t n for mesh adaptivity, for simplicity, we use the notation Thnonly for the last one, so the term u n−1hin (4) has not be re-interpolated here. The functionu n−1happears also in the estimator (10) and can be interpolated without great difficultyon the new elements of Th n . We introduce for all n = 1, ..., N, the spacial global errorestimators√ ∑˜η n =√ ∑√ ∑˜η n,K 2 , S n = Sn,K 2 , η n = ηn,K 2 . (13)K∈T n hK∈T n hK∈T n hWe also use the following notations : |τ| = max τ n, with each family of velocity1≤n≤N(v n ) 0≤n≤N , and pressure fields (Φ n ) 0≤n≤N , we agree to associate the affine functionsrespectively defined for all t ∈ [t n−1 , t n ] byv τ (t) = t − t n−1v n + t n − tv n−1 and Φ τ (t) = t − t n−1Φ n + t n − tτ nτ nτ nτ n(τn−1τ n)Φ n−1 ;similarly, we define v hτ and Φ hτ the affine functions respectively associated to (v n h ) 0≤n≤N,and (Φ n h ) 0≤n≤N; we set ẽ(t) = (u − ũ τ )(t), e(t) = (u − u τ )(t), ε(t) = (p − Φ τ )(t)and for all n = 1, ..., N, ẽ n = ũ n − ũ n h , en = u n − u n h , εn = Φ n − Φ n h .We use the following convention :(a ≼ b ⇐⇒ a ≤ c b) , (a ≃ b ⇐⇒ a ≼ b and a ≼ b) ,where the constant c must be independent of ν, τ and h.2.2. Statement of the main resultsWe intend to bound the errors u−ũ hτ , u−u hτ , and p−Φ hτ at each time-step t n as afunction of the error estimators. For any function field v, we use here, the decompositionv − v hτ = (v − v τ ) + (v τ − v hτ ), and we start (of course for some appropriate norms)by evaluating v − v τ . We don’t bring any proof of the estimates below.Proposition1 Assume the data f ∈ L 2 (0, T ; H), u 0 ∈ H and ν|τ| ≤ 1. Then, theTAMTAM –Tunis– 2005