By Eduard L. Stiefel

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Q. Chen et al. It is noted that Shape-GA1 operates for the left part and Shape-GA2 for the right part of the nozzle. In this paper, Shape-GA1 receives the y coordinate value and slope of the point D from the fittest curve s2 of Shape-GA2. This value will be used for the end control point of the B´ezier curve of s1 in ShapeGA1 for the next step. This treatment ensures continuity and is expected to have smoothness at the overlapping segment CD. Shape-GA2 does the same as Shape-GA1 meanwhile. The calculation of each shape fitness requires to solve the flow equations by CFD solvers over the whole domain.

For the continuous ﬁnite element spaces we show that the variable V-cycle provides an optimal preconditioner for the DG system. The piece-wise constant functions do not have approximation property so in order to control the energy growth of the inter-level transfer operator we apply W –cycle MG. Finally, we present a number of numerical experiments that support the theoretical ﬁndings. 1 Introduction Consider the following model second order elliptic problem on a bounded domain with a polygonal boundary Ω ⊂ Rd , d = 2, 3: −∇ · (a(x)∇u) = f (x) in Ω, u(x) = g on ∂Ω.

In the case of piece-wise constant coarse spaces (W -cycle) the advantage of Preconditioning of DG Methods 43 the weighted stabilization is evident – the numerical experiments show that the number of PCG iterations is essentially independent of the jumps. Table 5. Numerical results for Test Problem 2: SIPG with linear elements and stabilization parameter κE = κ {{a}}. Test Problem 2 DOF of SIPG precond. 001, V -cycle ǫ = 10−4 , V -cycle ǫ = 10−5 , V -cycle ǫ = 10−6 , V -cycle ǫ = 1, var. 1, var.