An adaptive finite element method for solving a double well problem describing crystalline microstructure

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.