Colloquium April 7
Date: Friday April 7
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Ronald H. W. Hoppe of Univ of Houston and Uni Augsburg
Title: Convergence Analysis of Adaptive Finite Element Methods for Constrained Distributed and Boundary Control Problems
Abstract
In this contribution, we are concerned with the development, analysis
and implementation of adaptive finite element methods for distributed
and boundary control problems with control constraints. The methods
presented in this contribution provide an error reduction and thus
guarantee convergence of the adaptive loop which consists of the essential
steps 'SOLVE', 'ESTIMATE', 'MARK', and 'REFINE'. Here, 'SOLVE'
stands for the efficient solution of the finite element discretized
problems.
The following step 'ESTIMATE' is devoted to a residual-type a
posteriori error estimation of the global discretization errors in the
state, the
co-state, the control and the co-control. A bulk criterion is the
core of
the step 'MARK' to indicate selected edges and elements for reønement,
whereas the final step 'REFINE' deals with the technical realization of
the refinement process itself.
The analysis is carried out for a model problem using discretizations
of the
state and the co-state by continuous, piecewise linear finite elements
and
of the control and the co-control by element-wise constants with respect
to a simplicial triangulation of the computational domain. Important
tools in the convergence proof are the reliability of the estimator,
a discrete local efficiency, and a perturbed Galerkin orthogonality.
Numerical results illustrate the performance of the error estimator.
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