Diploma thesis as pdf file - Johannes Kepler University, Linz

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3.3 Error estimate for Finite Element solution 31 Figure 3.2: The set K h is not in general contained in the set K. Let K h be approximation of the convex set K : K h = {v h ∈ V h |∀b ∈ N h : v h (b) ≥ ψ(b), v h = Π h u on Ω − Ω h }. Note that the set K h is not in general contained in the set K. Theorem 3.3.2 Assume that the solution u is in the space H 2 (Ω). Then the continuous piecewise linear approximation u h satisfies ‖u h − u‖ 1,Ω = O(h). Proof Using integration by parts and denoting by µ = −(∆u + f ), we find 〈Au − f ,v〉 = ∫ = − Ω∫ = 〈µ,v〉. Taking in (3.5) instead of v h = Π h u gives: ∇u∇vdω − 〈 f ,v〉 Ω ∆u vdω − 〈 f ,v〉 |u − u h | 1,Ω ≤ a(u − u h ,u − Π h u) + 〈µ,Π h u − u h 〉 Ωh , (3.8) since u h = Π h u on Ω − Ω h . From [7] the variational inequality (3.3) implies the following pointwise relations: µ ≥ 0 and µ(ψ − u) = 0 a.e. on Ω.
3.4 Numerical experiments 32 Hence 〈µ,Π h u − u h 〉 Ωh = 〈µ,Π h (u − ψ) − (u − ψ)〉 Ωh + 〈µ,u − ψ〉 + 〈µ,Π h ψ − u h 〉 Ωh ≤ 〈µ,Π h (u − ψ) − (u − ψ)〉 Ωh (3.9) ≤ ‖µ‖ L 2 (Ω h ) ‖Π h(u − ψ) − (u − ψ)‖ L 2 (Ω h ) = O(h2 ). (3.10) Here (3.9) was derived from the observation: from u h (b) ≥ Π h ψ(b) ∀b ∈ N h follows that u h ≥ Π h ψ on Ω h ; hence 〈µ,Π h ψ − u h 〉 Ωh ≤ 0. Thus, from (3.8) we can conclude: ‖u − u h ‖ 2 1,Ω ≤ C‖u − u h‖ 1,Ω ‖u − Π h u‖ 1,Ω + O(h 2 ). Taking into account that linear interpolate is of the first order approximation in H 1 , we obtain the error estimate ‖u − u h ‖ 1,Ω = O(h). 3.4 Numerical experiments In this section we present results for several examples. For all these examples the initial iterate was taken u (0) = 0. We know that the convergence rate of the solutions of the sequence of unconstrained minimization problems (UP) to the solution of the original problem is ‖u κ − ū‖ = O( √ κ). And accuracy of approximation by using piecewise finite elements will not be better than O(h). Consequently, we should relate the tolerance ε for κ with h in a proper way: e.g. ε = 0.1/N, where N - number of unknowns. Initial barrier parameter was chosen κ 0 = 1. ∥ For Newton iterations stopping criterion was ∥ u−ψ δu ∥ ≤ 0.5. L ∞ (Ω) For the integrations in computing of L 2 (Ω)-norms and checking of the conditions Ξ ≤ ( ) τ 2 C trapezoidal rule was used.
Page 1 and 2: ÆØÞÛÖĐÙÖÓÖ×ÙÒ¸ÄÖÙ
Page 3 and 4: Contents Acknowledgments iii Introd
Page 5 and 6: Acknowledgments My sincere gratitud
Page 7 and 8: Chapter 1 Mathematical modeling of
Page 9 and 10: 1.2 Existence and uniqueness of the
Page 11 and 12: 1.3 Alternative equivalent formulat
Page 17 and 18: 2.1 The central path 12 Problem 2.1
Page 19 and 20: 2.1 The central path 14 for some c
Page 21 and 22: 2.2 A predictor-corrector approach
Page 29 and 30: 3.1 Algorithm 24 Figure 3.1: Stayin
Page 31 and 32: 3.2 Finite element discretization 2
Page 33 and 34: 3.2 Finite element discretization 2
Page 35: 3.3 Error estimate for Finite Eleme
Page 39 and 40: 3.4 Numerical experiments 34 ations
Page 41 and 42: 3.4 Numerical experiments 36 Table
Page 43 and 44: 3.4 Numerical experiments 38 In the
Page 45 and 46: 3.4 Numerical experiments 40 3.4.4
Page 47 and 48: Conclusion In this work we consider
Page 49 and 50: Bibliography [1] Bonnans, J.F.; Gil
Page 51: Eidesstattliche Erklarung Ich, Serb

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