Xxxxxx Continuation Sample Clauses

Xxxxxx Continuation. The idea behind the continuation method is to have a non-stationary selection of ε, such that in the k-th iterate we have ε = ε(k) and limk→∞ ε(k) = 0. We start with a (large value of) ε and perform one Xxxxxx iteration. If this iteration reduces the residual by more than ε (in the algorithm below denoted by “success == true”), the new solution is accepted and the value of ε is reduced. If not (“success == false”), the new solution is discarded and ε is increased. The procedure continues until a specified target value of ε is reached. The resulting algorithm reads:

Xxxxxx Continuation εnew = ε0, εcurr = εnew, ρ = ρ0; u = u0, p = p0, W = 0; while εcurr > εtarget do [unew, pnew, Wnew, success]=Xxxxxx(u, p, W , f, εcurr); if sucess == true then εnew = εcurr, εcurr = ρεcurr, ρ = 0.9ρ; u = unew, p = pnew, W = Wnew; else ρ = 1+ρ ; εcurr = ρεnew; end if end while 4.3 Discretization There are different ways to discretize (4.1) in space, examples are the MAC dis- cretization on staggered grids and collocated finite difference methods [1, 66], finite volume [84] or LBB-stable finite elements [34]. This thesis will focus on Galerkin fi- nite element discretization methods. The application of the approach presented here to other discretization methods is however possible. Denote by Hh ⊂ H1, Qh ⊂ L2 s and Wh ⊂ L2 the finite dimensional subspaces for the velocity, pressure, and the auxiliary variable W , respectively. We assume throughout this thesis that the pair of spaces Hh and Qh is LBB stable [50]. The finite element method for (2.7) reads: Find uh ∈ Hh, ph ∈ Qh and Wh ∈ Wh such that a(uh, vh) + b(ph, vh) + c(vh, Wh) − b(qh, uh) − c(uh, Zh) + gε(uh, Wh, Zh) = (f, vh) (4.13) for any vh ∈ Hh, qh ∈ Qh and Zh ∈ Wh.