Adaptive algorithms for scalar non-convex variational problems

✍ Scribed by Carsten Carstensen; Petr Plecháč

• Book ID: 104308558
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 676 KB
• Volume: 26
• Category: Article
• ISSN: 0168-9274
• DOI: 10.1016/s0168-9274(97)00089-5

✦ Synopsis

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.