Adaptive algorithms for scalar non-convex variational problems

Abstract Since direct numerical solution of a non-convex variational problem (P) yields rapid oscillations, we study the relaxed problem (RP) which is a degenerate convex minimization problem. The classical example for such a relaxed variational problem is the double-well problem. In an earlier work, the authors showed that relaxation is not linked to a loss of information if our main interest concerns the macroscopic displacement field, the stress field or the microstructure. Furthermore, a priori and a posteriori error estimates have been computed and an adaptive algorithm was proposed for this class of degenerate variational problems. This paper addresses the question of efficiency and considers the ZZ-indicator, named after Zienkiewicz and Zhu, and discusses its performance compared with the rigorous indicator introduced by the authors.

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