

Sören Bartels: Reliable and efficient approximation of polyconvex envelopes
Nonconvex variational problems occur in the mathematical
modelling of phase transitions in crystalline solids. Since such
problems typically lack classical solutions relaxations are
considered to compute a generalised solution. This approach
and its numerical analysis are now well understood for scalar
problems for which the relaxed problem is (degenerately)
convex. In vectorial problems the quasiconvex envelope
of an energy density is needed to define the relaxed problem.
Such envelopes are only known for a few special cases. The
polyconvex envelope is a lower bound for the quasiconvex
envelope and the talk is devoted to the reliable and efficient
approximation of this lower bound. We present a quadratically
convergent iterative algorithm with grid coarsening and
local refinement to approximate polyconvex envelopes. The
algorithm allows for the simultaneous (polyconvex) relaxation
and approximation of certain vectorial variational problems and
this approach results in a numerical scheme with two discrete
scales.
