Tamtam Proceedings - lamsin

REE for Stokes equations 2031. Introduction and preliminariesLet Ω be a bounded connected domain of R d (d = 2, 3), with Lipschitz-continuousboundary Γ and we denote by ⃗n the unit outward normed vector to Ω on Γ. Let T > 0 bea real constant. We set X = H0 1 (Ω) d , Y = L 2 (Ω) d and M = H 1 (Ω)/R.For an initial velocity field, u 0 ∈ H = {v ∈ Y ; ∇.v = 0 in Ω , v.⃗n = 0 on Γ }, dataf ∈ L 2 (0, T ; H), and a given real kinematic viscosity ν > 0, we consider the incompressibletime-dependent Stokes problem :Find (u(t), p(t)) ∈ X × L 2 (Ω)/R such that for almost every t ∈]0, T [⎧d ⎪⎨ (u, v) + ν(∇u, ∇v) − (p, ∇.v) = (f, v) ∀ v ∈ X,dt(q, ∇.u) = 0 ∀ q ∈ L ⎪⎩2 (Ω)/R ,(1)u(0) = u 0 .Let us introduce a regular partition 0 = t 0 < t 1 < ... < t N = T of the time intervalτ n−1[0 , T ], with step sizes τ n = t n − t n−1 such that the regularity parameter max is1≤n≤N τ nbounded from above independently of τ. We denote by τ the N−tuplets (τ 1 , ..., τ N ) andby f n = 1 ∫ tnf(t)dt for each 1 ≤ n ≤ N. The variational form of the semi-discreteτ n t n−1Chorin-Temam projection scheme applied to the Stokes equations (1), reads :Find (ũ n , u n , Φ n ) 1≤n≤N ∈ (X × Y × M) N initialized by u 0 = u 0 and satisfying(ũ n , v) + ντ n (∇ũ n , ∇v) = (u n−1 , v) + τ n (f n , v) ∀v ∈ X, (2){ (u n , v) + τ n (∇Φ n , v) = (ũ n , v) ∀ v ∈ Y,(u n (3), ∇q) = 0 ∀ q ∈ M.By standard arguments, it is readily checked that problems (2) and (3) are well posed.Indeed step (2) constitutes an elliptic boundary value problem for an intermediate velocityunknown ũ n , which is a prediction of u(t n ) satisfying a homogeneous Dirichlet boundarycondition, but is not divergence free. The second step (3) represents a Darcy’s problemwhich determines the end-of-step divergence-free velocity u n which is a correction ofũ n , together with a suitable approximation of the pressure distribution Φ n . We easilycheck that Φ n is the solution of a Poisson problem with homogeneous Neumann boundaryconditions. We assume thar Ω is polygonal or polyhedral (d = 2 or 3). For each n,0 ≤ n ≤ N τ , we associate a regular triangulation Thn of Ω into triangles or tetrahedra. Foreach element K in Th n,we denote by h K the diameter of K, E K the set of edges or facesE of K wich are not contained in Γ and for each element E in E K we denote by h E thediameter of E. The fully discrete version of algorithm (2)-(3) in the framework of spatialGalerkin finite element approximation takes similar formulation written for the unknowsequencedenoted by (ũ n h , un h , Φn h ) 1≤n≤N τretrieved in some appropriate approximationsubspaces of X, Y and M.TAMTAM –Tunis– 2005