Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Preprint 2014-29

Carsten Carstensen and Georg Dolzmann


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


Abstract: The relaxation of the two-well model problem in the analysis of solid-solid phase transitions leads to a variational problem with a quasiconvex energy density which fails to be convex if the phases are not compatible. This paper presents an adaptive algorithm for the computation of minimizers for this functional in nite 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 degenerate convex part and a null-Lagrangian, some convexity control of the degenerate convex part, and some rened estimator reduction compatible with the translation energy.


Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2014-29


Subjclass: 65N12, 65N30


25 pp.