Numerical solution of Variational Inequalities by Adaptive Finite Elements

1. Front Matter

   Pages I-X

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2. Introduction

   Pages 1-12

3. Models in elasto-plasticity

   Pages 13-22

4. The dual-weighted-residual method

   Pages 23-32

5. Extensions to stabilised schemes

   Pages 33-45

6. Obstacle problem

   Pages 47-56

7. Signorini’s problem

   Pages 57-68

8. Strang’s problem

   Pages 69-73

9. General concept

   Pages 75-79

10. Lagrangian formalism

    Pages 81-89

11. Obstacle problem revisited

    Pages 91-94

12. Variational inequalities of second kind

    Pages 95-103

13. Time-dependent problems

    Pages 105-107

14. Applications

    Pages 109-127

15. Iterative Algorithms

    Pages 129-139

16. Conclusion

    Pages 141-141

17. Back Matter

    Pages 143-161

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This work describes a general approach to a posteriori error estimation and adaptive mesh design for ?nite element models where the solution is subjected to inequality constraints. This is an extension to variational inequalities of the so-called Dual-Weighted-Residual method (DWR method) which is based on a variational formulation of the problem and uses global duality arguments for deriving weighted a posteriori error estimates with respect to arbitrary functionals of the error. In these estimates local residuals of the computed solution are multiplied by sensitivity factors which are obtained from a - merically computed dual solution. The resulting local error indicators are used in a feed-back process for generating economical meshes which are tailored - cording to the particular goal of the computation. This method is developed here for several model problems. Based on these examples, a general concept is proposed, which provides a systematic way of adaptive error control for problems stated in form of variational inequalities. F¨ ur Alexandra, Katharina und Merle Contents 1 Introduction 1 2 Models in elasto-plasticity 13 2. 1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . 14 2. 2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 The dual-weighted-residual method 23 3. 1 A model situation in plasticity . . . . . . . . . . . . . . . . . . 24 3. 2 A posteriori error estimate . . . . . . . . . . . . . . . . . . . . . 25 3. 3 Evaluation of a posteriori error bounds . . . . . . . . . . . . . . 26 3. 4 Strategies for mesh adaptation . . . . . . . . . . . . . . . . . . 28 3. 5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Extensions to stabilised schemes 33 4. 1 Discretisation for themembrane-problem . . . . . . . . . . . . 35 4. 2 A posteriori error analysis . . . . . . . . . . . . . . . . . . . . . 37 4. 3 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . .

• A Posteriori Fehlerschätzung
• A Priori Fehlerschätzung
• Adaptivität
• Finite
• Finite-Elemente-Methode
• Variationsungleichungen
• equation
• finite element method
• mathematics

Dr. Franz-Theo Suttmeier is a professor of Scientific Computing at the Institute of Applied Analysis and Numerics at the University of Siegen.

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