A comparison of classical and new finite element methods for the computation of laminate microstructure Original Research Article

A geometrically nonlinear continuum theory has been developed for the equilibria of martensitic crystals based on elastic energy minimization. For these non-convex functionals, typically no classical solutions exist, and minimizing sequences involving Young measures are studied. Direct minimizations using discretization based on conforming, non-conforming, and discontinuous elements have been proposed for the numerical approximation of this problem. Theoretical results predict the superiority of the discontinuous finite element. Detailed numerical studies of the available finite element discretizations in this paper validate the theory. One-dimensional prototype problems due to Bolza and Tartar and a two-dimensional numerical model of the Ericksen–James energy are presented. Both classical elements yield solutions that possess suboptimal convergence rates and depend heavily on the underlying numerical mesh. The discontinuous finite element method overcomes this problem and shows optimal convergence behavior independent of the numerical mesh.