Strong Convergence in Stabilised Degenerate Convex Problems

Solutions to non–convex variational problems typically exhibit enforced finer and finer oscillations called microstructures such that the infimal energy is not attained. Those oscillations are physically meaningful, but finite element approximations typically experience dramatic difficulty in their reproduction. The relaxation of the non–convex minimisation problem by (semi–)convexification leads to a macroscopic model for the effective energy. The resulting discrete macroscopic problem is degenerate in the sense that it is convex but not strictly convex. This paper discusses a modified discretisation by adding a stabilisation term to the discrete energy. It will be announced that, for a wide class of problems, this stabilisation technique leads to strong H¹–convergence of the macroscipic variables even on unstructured triangulations. This is in contrast to the work 2] for quasi–uniform triangulations and enables the use of adaptive algorithms for the stabilised formulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)