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

The minimization of nonconvex functionals naturally arises in materials sci ences 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 obser ved in experiments [2, 3]. - From the numerical point of view, classical co nforming and nonconforming finite element discretizations have been observe d to give minimizers with their quality being highly dependent on the under lying triangulation, see [8, 24, 26, 27] for a survey. Recently, a new appr oach has been proposed and analyzed in [15, 16] that is based on discontinu ous finite elements to reduce the pollution effect of a general triangulati on on the computed minimizer. The goal of the present paper is to propose a nd analyze an adaptive method, giving a more accurate resolution of laminat ed microstructure on arbitrary grids.