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Description of the Lecture Series

The purpose of the keynote lectures is to give an overview over current a posteriori error estimation techniques, related mesh adaptation strategies and their application to problems in Computational Science and Engineering. They will be targeted at an audience with basic knowledge of the finite element method, as for instance the key chapters of the books by Braess, Brenner and Scott, Carey and Oden, Ciarlet or Strang and Fix. The topics will cover material on adaptivity and a posteriori error estimation (see Bangerth and Rannacher). The ten lectures of this conference will center around the following areas:
  1. Basic concepts: Variational form of partial differential equations; finite element approximation; a priori versus a posteriori error analysis; Galerkin orthogonality and duality arguments; residual-based a posteriori error estimates; reliability and efficiency of error estimates.
  2. Adaptive mesh refinement strategies: Local error indicators; strategies for mesh adaptation; hanging nodes and transition elements; optimal mesh refinement; feature-based versus residual-based mesh adaptation.
  3. Goal-oriented error estimation: The dual weighted residual (DWR) method; examples of functional estimates; evaluation of error estimators; practical aspects; illustrative examples.
  4. Time-dependent problems: The DG method for time discretization; the nonstationary dual problem; efficient computation of dual solutions; evaluation of error estimators; sample applications: heat and wave equation.
  5. Nonlinear problems: Derivation of estimates for nonlinear problems; augmentation of nonlinear iteration for the dual problem; application to the incompressible Navier-Stokes equations.
  6. Applications I: Application of the DWR technique to technical flow problems; Euler and Navier-Stokes equations; numerical stabilization; drag and lift computation.
  7. Applications II: Application of the DWR technique to coupled systems; chemically reactive flows; fluid-structure interaction problems.
  8. Eigenvalue problems: Finite element approximation of eigenvalue problems; the eigenvalue problems as a coupled nonlinear system; a priori and a posteriori error estimates for eigenvalues; application to stability analysis of stationary flow.
  9. Optimization: Optimal control and parameter estimation problems; derivation of the dual problem; tracking-type problems; discrete parameter estimation; sample applications.
  10. Applications III: Parameter estimation and model calibration for chemically reactive flow; problems with control and state-constraints.

Back to 2009 CBMS Conference on Adaptive Finite Element Methods.