Diploma thesis as pdf file - Johannes Kepler University, Linz

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1.2 Existence and uniqueness of the solution 3 here and in the following µ stays for the Lebesgue measure. Then the total energy is: J(v) = 1 2 ∫ Ω ∫ | ∇v | 2 dω − f vdω. Ω The ”obstacle problem” consists in finding the equilibrium state of the membrane, i.e. in minimizing the energy functional J(v), when the deflection of the membrane is restricted from below by the obstacle. Then the set of admissible deflections is given as : K = {v ∈ H 1 0 (Ω)|v ≥ ψ a.e. in Ω}. We see that the K is not a linear set. Throughout this work we assume that ψ ∈ L 2 (Ω) and f ∈ H −1 (Ω). Thus, we come up with the following: Problem 1.1.1 Given a bounded domain Ω ⊂ R 2 and functions f ∈ H −1 andψ ∈ L 2 (Ω), find a solution u ∈ H 1 0 such that J(u) = minJ(v) ∀v ∈ K, v∈K where the functional J(v) : H0 1 → R is represented by : J(v) = 1 ∫ |∇v| 2 dω − 〈 f ,v〉. 2 1.2 Existence and uniqueness of the solution Ω For the existence and uniqueness of the solution of the problem (1.1.1) we bring here the following statement for more general problem: Theorem 1.2.1 Let H be a Hilbert space, K ⊂ H be closed and convex, the continuous bilinear form a(·,·) : H × H → R be symmetric and coercive, i.e. ∃α > 0 : a(v,v) ≥ α‖v‖ 2 v ∈ H,
1.2 Existence and uniqueness of the solution 4 and f ∈ H ∗ . Then there exists unique solution to the minimization problem: to find u ∈ K : J(u) = minJ(v) ∀v ∈ K, v∈K where the functional J : H → R is defined by J(v) = 1 a(v,v) − 〈 f ,v〉. 2 Remark Here 〈 f ,v〉 is the pairing between f and v, i.e. 〈 f ,v〉 = f (v). Definition 1.2.2 The point y ∈ K such that ‖x − y‖ ≤ ‖x − z‖ ∀z ∈ K is called the projection of x onto K. For the proof of the theorem we need the following lemma: Lemma 1.2.3 If K is closed and convex subset of a Hilbert space then each x ∈ H admits unique projection on K. Proof of the lemma (1.2.3) Let d = inf ‖x − z‖. Then we can find a sequence z∈K {η k } ∈ K : lim ‖η k − x‖ = d. Since K is convex, for any η m ,η n ∈ {η k } it holds k→∞ 1 2 (η m + η n ) ∈ K and d 2 ∥ ≤ ∥x − 1 2 (η m + η n ) ∥ 2 . Applying the parallelogram law for Hilbert space: 2‖x − η m ‖ 2 + 2‖x − η n ‖ 2 = ‖η n − η m ‖ 2 ∥ + 2∥x − 1 2 (η n + η m ) ∥ 2 . Hence: ‖η n − η m ‖ 2 ≤ 2‖x − η m ‖ 2 + 2‖x − η n ‖ 2 − 4d 2 .
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: Chapter 1 Mathematical modeling of
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 and 36: 3.3 Error estimate for Finite Eleme
Page 37 and 38: 3.4 Numerical experiments 32 Hence
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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