The minimization of nonconvex functionals naturally arises in
materials sciences where deformation gradients in certain alloys exhibit
microstructures. For example, minimizing sequences of the nonconvex
Ericksen-James energy can be associated with deformations in
martensitic materials that
are observed in experiments[2,3].
— From the numerical
point of view, classical conforming and nonconforming finite element
discretizations have been observed to give minimizers
with their quality being highly
dependent on the underlying triangulation, see [8,24,26,27] for
a survey. Recently, a new
approach has been proposed and analyzed in [15,16]
that is
based on discontinuous finite elements to reduce the pollution effect
of a general triangulation on the computed minimizer.
The goal of the present paper is
to propose and analyze
an adaptive method,
giving a more accurate resolution of laminated microstructure
on arbitrary grids.