Convergence of adaptive finite element methods for a nonconvex double-well minimization problem

This paper focuses on the numerical analysis of a nonconvex variational problem which is related to the relaxation of the two-well problem in the analysis of solid-solid phase transitions with incompatible wells and dependence on the linear strain in two dimensions. The proposed approach is based on the search for minimizers for this functional in finite element spaces with Courant elements and with successive loops of the form SOLVE, ESTIMATE, MARK, and REFINE. Convergence of the total energy of the approximating deformations and strong convergence of all except one component of the corresponding deformation gradients is established. The proof relies on the decomposition of the energy density into a convex part and a null-Lagrangian. The key ingredient is the fact that the convex part satisfies a convexity property which is stronger than degenerate convexity and weaker than uniform convexity. Moreover, an estimator reduction property for the stresses associated to the convex part in the energy is established.