convergence
adaptive algorithm
optimal design problem
A convergent adaptive finite element method for an optimal design problem
Soeren Bartels
Bartels
Soeren
Carsten Carstensen
Carstensen
Carsten
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 29 pages, figures
A convergent adaptive finite element method for an optimal design problem
Soeren Bartels
,
Carsten Carstensen
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 29 pages, figures
MSC 2000
 65N12 Stability and convergence of numerical methods

65N30 Finite elements, RayleighRitz and Galerkin methods, finite methods
Abstract
Abstract. The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is
one model example in topology optimisation. It eventually leads to a degenerated convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables $u$ but unique derivatives $\sigma := DW(Du)$. Even sharp a posteriori error estimates still suffer from the reliabilityefficiency gap. However, it motivates a simple edgebased adaptive meshrefining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined
convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed adaptive finite element method (AFEM).
This document is wellformed XML.