Adaptive Finite Element Methods for Differential Equations

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 1-14

3. An ODE Model Case

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 15-24

4. A PDE Model Case

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 25-40

5. Practical Aspects

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 41-60

6. The Limits of Theoretical Analysis

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 61-70

7. An Abstract Approach for Nonlinear Problems

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 71-80

8. Eigenvalue Problems

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 81-100

9. Optimization Problems

   • Wolfgang Bangerth, Rolf Rannacher

   Pages 101-112

10. Time-Dependent Problems

    • Wolfgang Bangerth, Rolf Rannacher

    Pages 113-128

11. Applications in Structural Mechanics

    • Wolfgang Bangerth, Rolf Rannacher

    Pages 129-142

12. Applications in Fluid Mechanics

    • Wolfgang Bangerth, Rolf Rannacher

    Pages 143-160

13. Miscellaneous and Open Problems

    • Wolfgang Bangerth, Rolf Rannacher

    Pages 161-165

14. Back Matter

    Pages 167-208

    PDF

These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self­ adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er­ ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low­ Mach-number flows with chemical reactions.

• Differential equations
• differential equation
• eigenvalue
• fluid mechanics
• mechanics
• numerical analysis
• solution
• ordinary differential equations

"Most graduate students in engineering and physical sciences should be able to handle the material without excessive difficulty. The presentation is very much a tutorial approach promoting a hands-on experience, reinforced with practical exercises at the end of each chapter, aimed towards practitioners.... [The] present book provides a gentler introduction for the beginning graduate student or nonspecialist practitioner."

— SIAM Review 

 

• TICAM, The University of Texas at Austin, Austin, USA

  Wolfgang Bangerth

• Institute of Applied Mathematics, University of Heidelberg, Heidelberg, Germany

  Rolf Rannacher

Policies and ethics