A convergent adaptive finite element method for an optimal design problem

Soeren Bartels , Carsten Carstensen

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 29 pages, figures

MSC 2000

65N12 Stability and convergence of numerical methods

65N30 Finite elements, Rayleigh-Ritz 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 reliability-efficiency gap. However, it motivates a simple edgebased adaptive mesh-refining 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).