ADAPTIVE FINITE ELEMENTS FOR RELAXED METHODS
(FERM) IN COMPUTATIONAL MICROSTRUCTURES
CARSTEN CARSTENSEN
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 ele-
ment 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.
Institute for Applied Mathematics and Numerical Analysis, Vienna University of
Technology, Wiedner Hauptstraße 8-10/115, A-1040 Vienna, Austria
E-mail address: carsten.carstensen@tuwien.ac.at
1991 Mathematics Subject Classification. 65N30, 65R20, 73C50.
Key words and phrases. computational microstructure, relaxation theory, nonconvex mini-
mization, phase transition.
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