The minimization of nonconvex functionals naturally arises inmaterials sciences where deformation gradients in certain alloys exhibitmicrostructures. For example, minimizing sequences of the nonconvexEricksen-James energy can be associated with deformations in martensitic materials thatare observed in experiments[2,3]. — From the numericalpoint of view, classical conforming and nonconforming finite elementdiscretizations have been observed to give minimizerswith their quality being highlydependent on the underlying triangulation, see [8,24,26,27] for a survey. Recently, a newapproach has been proposed and analyzed in [15,16]that is based on discontinuous finite elements to reduce the pollution effectof a general triangulation on the computed minimizer.The goal of the present paper isto propose and analyzean adaptive method,giving a more accurate resolution of laminated microstructure on arbitrary grids.