Seminar series
Date
Mon, 23 Feb 2004
17:00
17:00
Location
L1
Speaker
Carsten Carstensen
Organisation
Bristol
Nonconvex minimisation problems are encountered in many applications such
as phase transitions in solids (1) or liquids but also in optimal design
tasks (2) or micromagnetism (3). In contrast to rubber-type elastic materials
and many other variational problems in continuum mechanics, the minimal
energy may be not attained. In the sense of (Sobolev) functions, the
non-rank-one convex minimisation problem (M) is ill-posed: As
illustrated in the introduction of FERM, the gradients of infimising
sequences are enforced to develop finer and finer oscillations called
microstructures. Some macroscopic or effective quantities, however, are
well-posed and the target of an efficient numerical treatment.
The presentation proposes adaptive mesh-refining algorithms for the finite
element method for the effective equations (R), i.e. the macroscopic
problem obtained from relaxation theory. For some class of convexified model
problems, a~priori and a~posteriori error control is available with an
reliability-efficiency gap. Nevertheless, convergence of some adaptive finite
element schemes is guaranteed. Applications involve model situations for (1),
(2), and (3) where the relaxation is provided by a simple
convexification.