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Adaptive Finite Element Methods for Optimization in Partial Differential Equations

Kapp, Hartmut Ulrich

German Title: Adaptive Finite Elemente Methoden für Optimierung mit Partiellen Differentialgleichungen

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Translation of abstract (English)

A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by (elliptic) partial differential equations. The Lagrangian formalism yields the first-order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive Galerkin finite element method. The mesh design in the resulting reduced models is controlled by residual-based a posteriori error estimates. These are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error. In this case, the computed state and co-state variables can be used as sensitivity factors multiplying the local cell-residuals in the error estimators. This results in a generic and efficient algorithm for mesh adaptation within the optimization process. Applications of the developed method are boundary control problem models governed by Ginzburg-Landau equations (superconductivity in semi-conductors), by Navier-Stokes equations, and by the Boussinesq viscosity model (flow with temperature transport for zero gravitation). Computations with more than 2 million unknowns were performed.

Document type: Dissertation
Supervisor: Rannacher, Prof. Dr. Rolf,
Date of thesis defense: 13 December 2001
Date Deposited: 29 Jan 2002 00:00
Date: 2000
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Institut für Mathematik
DDC-classification: 510 Mathematics
Controlled Keywords: Nichtlineare Optimierung, Optimierung / Nebenbedingung, Nichtlineare partielle Differentialgleichung, Finite-Elemente-Methode
Uncontrolled Keywords: A Posteriori Fehlerschätzer , adaptive Gitterverfeinerung , Ginzburg-Landau Modell , Navier-Stokes Gleichungen , Boussinesq Modelloptimal control problem , finite elements , a posteriori error estimates , mesh adaptation , Navier-Stokes equations
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