## Adaptive Finite Element Methods for Differential EquationsThese 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. |

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### Contents

Introduction | 1 |

computation of drag coefficient | 2 |

12 The need for goaloriented mesh adaptation | 4 |

13 Further examples of goaloriented simulation | 9 |

14 General concepts of error estimation | 11 |

An ODE Model Case | 15 |

FD versus FE method | 19 |

23 Exercises | 23 |

71 A posteriori error analysis | 82 |

72 Error control for functionals of eigenfunctions | 91 |

73 The stability eigenvalue problem | 95 |

74 Exercises | 99 |

Optimization Problems | 101 |

81 A posteriori error analysis via Lagrange formalism | 103 |

82 Application to a boundary control problem | 105 |

83 Application to parameter estimation | 110 |

A PDE Model Case | 25 |

31 Finite element approximation | 26 |

32 Global a posteriori error estimates | 29 |

33 A posteriori error estimates for output functionals | 30 |

34 Higherorder finite elements | 37 |

35 Exercises | 39 |

Practical Aspects | 41 |

41 Evaluation of the error identity and indicators | 42 |

42 Mesh adaptation | 46 |

43 Use of error estimators for postprocessing | 52 |

44 Towards anisotropic mesh adaptation | 55 |

45 Exercises | 60 |

The Limits of Theoretical Analysis | 61 |

51 Convergence of residuals | 64 |

52 Approximation of weights | 65 |

53 Exercises | 69 |

An Abstract Approach for Nonlinear Problems | 71 |

61 Galerkin approximation of nonlinear equations | 72 |

62 A nested solution approach | 78 |

63 Exercises | 79 |

Eigenvalue Problems | 81 |

84 Exercises | 111 |

TimeDependent Problems | 113 |

the heat equation | 115 |

the wave equation | 123 |

94 Exercises | 128 |

Applications in Structural Mechanics | 129 |

102 A model problem in elastoplasticity theory | 134 |

103 Exercises | 142 |

Applications in Fluid Mechanics | 143 |

111 Computation of drag and lift in a viscous flow | 144 |

112 Minimization of drag by boundary control | 152 |

113 Stability analysis for stationary flow | 156 |

114 Exercises | 160 |

Miscellaneous and Open Problems | 161 |

122 Current developments | 162 |

123 Open problems | 164 |

Solutions of exercises | 167 |

Bibliography | 191 |

203 | |

### Other editions - View all

Adaptive Finite Element Methods for Differential Equations Wolfgang Bangerth,Rolf Rannacher Limited preview - 2013 |

Adaptive Finite Element Methods for Differential Equations Wolfgang Bangerth,Rolf Rannacher No preview available - 2014 |

### Common terms and phrases

accuracy adapted meshes adjoint anisotropic approximate error Babuska base solution Becker bilinear biquadratic boundary Braack cell-wise Chapter computed consider cost functional defined derivative discrete dual domain drag coefficient dual problem dual solution DWR method edge residuals eigenfunctions eigenvalue problem energy-norm error estimator error functional error indicators evaluation example Exercise 2.1 finite element approximation finite element method flow Galerkin approximation Galerkin method Galerkin orthogonality global refinement hanging nodes heuristic Heuveline and Rannacher higher-order interpolation ISBN iteration J(uh K€Th KeTh L2-norm L2-norm error Lagrangian linear mesh adaptation mesh refinement model problem Navier-Stokes equations norm numerical obtain optimal mesh output functional parameter piecewise Poisson problem post-processing posteriori error analysis posteriori error estimate posteriori error representation primal and dual Proposition refined meshes remainder term Remark Ritz projection semilinear form solving space space-time stability constant stability eigenvalue problem stationary points strategy Suttmeier variational formulation weighted error estimator

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