Adaptive algorithms for scalar non-convex variational problems

Applied Numerical Mathematics

Volumn 26, Issue 1-2, 1998, Pages 203-216

  Carstensen, Carsten ^a   Plecháč, Petr ^b  

^a UNIVERSITY OF KIEL   (Germany)

^b HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

A posteriori error estimates; Adaptive algorithms; Microstructure; Non convex minimization; Young measures; Zz estimator

Indexed keywords

ERROR ANALYSIS; ESTIMATION; OPTIMIZATION; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

A POSTERIORI ERROR ESTIMATION; NON-CONVEX MINIMIZATION; NON-CONVEX VARIATIONAL PROBLEMS; ZIENKIEWICZ-ZHU ESTIMATOR;

ADAPTIVE ALGORITHMS;

EID: 0031644439     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(97)00089-5     Document Type: Article

References (17)

1. J.M. Ball, A version of the fundamental theorem for Young measures, in: M. Rascle, D. Serre and M. Slemrod, eds., Partial Differential Equations and Continuum Models of Phase Transitions, Lecture Notes in Physics 344 (Springer, Berlin, 1989) 207-215.
2. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100 (1987) 13-52.
3. J.M. Ball and R.D. James, Proposed experimental tests of the theory of fine microstructure and the two-well problem, Phil. Trans. Roy Soc. Lond. Set A 338 (1992) 389-450.
4. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15 (Springer, New York, 1994).
5. C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp. (1997).
6. M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells, SIAM J. Numer. Anal. 29 (1992) 1002-1019.
7. C. Collins and M. Luskin, Optimal order error estimates for the finite element approximation of a solution of a non-convex variational problem, Math. Comp. 57 (1991) 621-637.
8. B. Dacorogna, Direct Methods in the Calculus of Variations (Springer, Berlin, 1989).
9. D. French, On the convergence of finite element approximations of a relaxed variational problem, SIAM J. Numer. Anal. 27 (1990) 419-436.
10. G. Friesecke, A necessary and sufficient condition for non-attainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Roy Soc. Edinburgh Sect. A 124 (1994) 437-471.
11. J. Kristensen, On the non-locality of quasiconvexity, Annal. Inst. H. Poincaré, Anal. Non-lineare, to appear.
12. M. Luskin, On the computation of crystalline micro structure, Acta Numer. 5 (1996) 191-257.
13. S. Nash, Newton-type minimization via the Lanczos method, SIAM J. Numer. Anal. 21 (1984) 770-787.
14. R.A. Nicolaides and N.J. Walkington, Computation of microstructure utilizing Young measure representations, in: C.A. Rogers and R.A. Rogers, eds., Recent Advances in Adaptive and Sensory Materials and their Applications (Technomic Publishing Co., Lancaster, 1992) 131-141.
15. R.A. Nicolaides and N.J. Walkington, Strong convergence of numerical solutions to degenerate variational problems, Math. Comp. 64 (1995) 117-127.
16. T. Roubíček, Relaxation in Optimization Theory and Variational Calculus (Walter de Gruyter, Berlin, 1996).
17. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretization of elliptic equations, Math. Comp. 62 (1994) 445-475.