On the Strong Convergence of Gradients in Stabilized Degenerate Convex Minimization Problems

Infimizing sequences in nonconvex variational problems typically exhibit enforced finer and finer oscillations called microstructures such that the infimal energy is not attained. Although those oscillations are physically meaningful, finite element approximations experience difficulty in their reconstruction. The relaxation of the nonconvex minimization 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 studies a modified discretization by adding a stabilization term to the discrete energy. It will be proven that for a wide class of problems, this stabilization technique leads to strong $H^1$ convergence of the macroscopic variables even on unstructured triangulations. In contrast to the work [C. Carstensen, P. Plechacd, S. Bartels, and A. Prohl, Interfaces Free Bound., 6 (2004), pp. 253-269] on quasi-uniform triangulations, this paper allows for general unstructured shape-regular triangulations and so enables the use of adaptive algorithms for the stabilized formulations.