Adaptive approximation of young measure solutions in scalar nonconvex variational problems

SIAM Journal on Numerical Analysis

Volumn 42, Issue 2, 2004, Pages 505-530

  Bartels, Sören ^a  

^a University of Maryland   (United States)

Author keywords

Adaptivity; Error estimation; Finite elements; Microstructure; Multiple scales; Nonconvex variational problems

Indexed keywords

ADAPTIVITY; ERROR ESTIMATION; FINITE ELEMENTS; MULTIPLE SCALES; NONCONVEX VARIATIONAL PROBLEMS;

APPROXIMATION THEORY; FINITE ELEMENT METHOD; INTEGRATION; MATHEMATICAL MODELS; MICROSTRUCTURE; PARAMETER ESTIMATION; PHASE TRANSITIONS; SET THEORY;

PROBLEM SOLVING;

EID: 10044262387     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902404091     Document Type: Article

References (30)

1. J.M. BALL, A version of the fundamental theorem for Young measures, in Partial Differential Equations and Continuum Models of Phase Transitions, M. Rascle, D. Serre, and M. Slemrod, eds., Lecture Notes in Phys. 344, Springer, Berlin, 1989, pp. 207-215.
2. J.M. BALL, P.J. HOLMES, R.D. JAMES, R.L. PEGO, AND P.J. SWART, On the dynamics affine structure, J. Nonlinear Sci., 1 (1991), pp. 17-70.
3. S. BARTELS, C. CARSTENSEN, AND G. DOLZMANN, Inhomogeneous Dirichlet Conditions in A Priori and A Posteriori Finite Element Error Analysis, Berichtsreihe des Mathematischen Seminars Kiel, Technical report, 02-1, Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 2001.
4. C. CARSTENSEN, Quasi interpolation and a posteriori error analysis in finite element method, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 1187-1202.
5. C. CARSTENSEN AND S. BARTELS, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids, part I: Low order conforming, nonconfarming, and mixed FEM, Math. Comp., 71 (2002), pp. 945-969.
6. C. CARSTENSEN AND S. MÜLLER, Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., 34 (2002), pp. 495-509.
7. C. CARSTENSEN AND P. PLECHÁČ, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp., 66 (1997), pp. 997-1026.
8. C. CARSTENSEN AND T. RouBÍǑEK, Numerical approximation of Young measures in non-convex variational problems, Numer. Math., 84 (2000), pp. 395-414.
9. M. CHIPOT AND S. MÜLLER, Sharp energy estimates for finite element approximations of nonconvex problems, in Variations of Domain and Free-Boundary Problems in Solid Mechanics, Solid Mech. Appl., 66, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997, pp. 317-325.
10. P.G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
11. F.H. CLARKE, Optimization and Nonsmooth Analysis, Classics Appl. Math. 5, SIAM, Philadelphia, 1990.
12. C. COLLINS AND M. LuSKiN, Optimal order error estimates for the finite element approximation of the solution of a non-convex variational problem, Math. Comp., 57 (1991), pp. 621-637.
13. J.L. ERICKSEN, Constitutive theory for some constrained elastic crystals, J. Solids and Structures, 22 (1986), pp. 951-964.
14. 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), pp. 437-471.
15. J. GONDZIO, HOPDM (version 2.12,) - A fast LP solver based on a primal-dual interior point method, European J. Oper. Res., 85 (1995), pp. 221-225.
16. D. KINDERLEHRER AND P. PEDREGAL, Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal., 23 (1991), pp. 1-19.
17. M. KRUŽÍK, Numerical approach to double well problems, SIAM J. Numer. Anal., 35 (1998), pp. 1833-1849.
18. M. KRUŽÍK AND A. PROHL, Young measure approximation in micromagnetism, Numer. Math., 90 (1998), pp. 291-307.
19. K. KURATOWSKI, Topology I, Academic Press, New York, PWN, Warszawa, 1966.
20. K. KURATOWSKI, Topology II, Academic Press, New York, PWN, Warszawa, 1968.
21. M. LUSKIN, On the computation of crystalline microstructure, Acta Numerica, 5 (1996), pp. 191-257.
22. A.-M. MATACHÉ, T. ROUBÍČEK, AND C. SCHWAB, Higher-order convex approximations of Young measures in optimal control, Adv. Comput. Math., 19 (2003), pp. 73-97.
23. S. MÜLLER, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems, Lecture Notes in Math. 1713, S. Hildebrandt and M. Struwer, eds., Springer, Berlin, 1999, pp. 85-210.
24. R.-A. NICOLAIDES AND N.J. WALKINGTON, Computation of microstructure utilizing Young measure representations, in Transactions of the Tenth Army Conference on Applied Mathematics and Computing, U.S. Army Research Office, Research Triangle Park, NC, 1993, pp. 57-68.
25. P. PEDREGAL, Numerical approximation of parametrized measures, Numer. Funct. Anal. Optim., 16 (1995), pp. 1049-1066.
26. P. PEDREGAL, Parametrized Measures and Variational Principles, Birkhäuser, Basel, 1997.
27. T. ROUBÍČEK, Relaxation in Optimization Theory and Variational Calculus, De Gruyter, New York, 1997.
28. T. ROUBÍČEK AND M. KRUŽÍK, Adaptive approximation algorithm for relaxed optimization problems, in Fast Solution of Discretized Optimization Problems, K.-H. Hoffmann, R. H. W. Hoppe, and V. Schulz, eds., Birkhäuser, Basel, 2001, pp. 242-254.
29. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Stuttgart, 1996.
30. L.C. YOUNG, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Comptes Rendues de la Société des Sciences et des Lettres de Varsovie, classe III, 30 (1937), pp. 212-234.