Diploma thesis as pdf file - Johannes Kepler University, Linz

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3.3 Error estimate for Finite Element solution 29 3.2.2 Predictor step At the predictor step we need to solve for p the following variational problem: Writing explicitly, find p ∈ V : ∫ Ω J ′′ κ(u + )(p,v) = −J ′ (u + )v ∀v ∈ V. ∫ ∇p∇vdω + κ Ω pv (u + − ψ) ∫Ω 2 dω = − ∫ ∇u + ∇vdω + f vdω. Ω Discretization by using piecewise linear functions leads to the system: A + p h = b + , where A + ∈ R N×N : V h → R N has components: A + i j = ∫ b + ∈ R N with components b + i Ω = −∑u + j j and p h ∈ R N with components p j . ∫ φ i φ j ∇φ i ∇φ j dω + κ Ω (u + h − , ψ)2 ∫ Ω ∫ ∇φ i ∇φ j dω + f φ i dω, Ω 3.3 Error estimate for Finite Element solution First we bring the theorem about the error estimate for the approximation which is valid for a general class of approximations schemes for variational inequalities. We consider in the Hilbert space V the problem : find u ∈ K : a(u,v − u) ≥ 〈 f ,v − u〉 ∀v ∈ K, (3.3) where K ⊂ V is a convex set, a(·,·) is continuous bilinear form, f ∈ V ∗ .
3.3 Error estimate for Finite Element solution 30 Let V h be a finite-dimensional subspace of the space V. We replace K by a convex set K h ⊂ V h . The corresponding discrete problem is find u h ∈ K h : a(u h ,v h − u h ) ≥ 〈 f ,v h − u h 〉 ∀v h ∈ K h , (3.4) We define the linear mapping : A : V → V ∗ such that 〈Au,v〉 = a(u,v) ∀v ∈ V. Theorem 3.3.1 If u and u h satisfy (3.3) and (3.4) respectively, then a(u − u h ,u − u h ) ≤ a(u − u h ,u − v h ) + 〈Au − f ,v h − u h 〉 ∀v h ∈ K h (3.5) a(u − u h ,u − u h ) ≤ a(u − u h ,u − v h ) + 〈Au − f ,v h − u〉 + 〈Au − f ,v − u h 〉 (3.6) v h ∈ K h , ∀v ∈ K Proof We have a(u − u h ,u − u h ) = a(u − u h ,u − v h ) + a(u − u h ,v h − u h ) = a(u − u h ,u − v h ) + 〈Au − f ,v h − u h 〉 + 〈 f ,v h − u h 〉 − a(u h ,v h − u h ) (from (3.4)) ≤ a(u − u h ,u − v h ) + 〈Au − f ,v h − u h 〉. (3.7) (3.3) implies 〈Au − f ,v − u〉 ≥ 0 ∀v ∈ K, then 〈Au − f ,v h − u h 〉 = 〈Au − f ,v h − u〉 + 〈Au − f ,u − v〉 + 〈Au − f ,v − u h 〉 ≤ 〈Au − f ,v h − u〉 + 〈Au − f ,v − u h 〉. From this and (3.7) follows (3.6). Now, we replace Ω with its polygonal approximation Ω h . We choose shape regular triangulation of Ω h such that all the vertices of T h which are on the boundary of the set Ω h are also on ∂Ω. Let h be triangulation parameter, i.e. the diameter of the biggest triangle is less than h. With such a triangulation T h we associate subspace V h ⊂ H0 1 (Ω) with piecewise linear elements defined by function values at the triangle vertices. We denote by Π h u piecewise linear interpolate of u.
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: 3.2 Finite element discretization 2
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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