Some remarks on nonuniqueness of minimizers for discrete minimization problems

• Autorzy: Chipot, M. , Elfanni, A.
• Języki publikacji: EN

Abstrakty

EN

We investigate the issue of uniqueness and nonuniqueness of minimizers for the approximation of variational problems. We show that when the continuous problem does not admit a minimizer its approximation by finite elements may lead to several discrete minimizers.

Słowa kluczowe

PL

przybliżenie; nie wypukły; rachunek wariacyjny; minimalizować; elementy skończone

EN

approximation; nonconvex; calculus of variations; minimizers; finite elements

• Czasopismo: Journal of Applied Analysis
• Rocznik: 2001
• Tom: Vol. 7, nr 2
• Strony: 191-208
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Chipot, M.

• Institut für Mathematik Universität Zürich Winterthurerstr. 190 CH-8057 Zürich, Switzerland

autor

Elfanni, A.

• Institut für Mathematik Universität Zürich Winterthurerstr. 190 CH-8057 Zürich, Switzerland

Bibliografia

• [1] Brighi, B., Sur quelques problèmes de calcul des variations et l’approximation de leur fonctionnelles relaxées, Thesis, University of Metz, 1991.
• [2] Brighi, B. and Chipot, M., Approximated convex envelope of a function, SIAM J. Numer. Anal. 31(1) (1994), 128-148.
• [3] Chipot, M., Numerical analysis of oscillations in nonconvex problems, Numer. Math. 59 (1991), 747-767.
• [4] Chipot, M., The appearence of microstructures in problems with incompatible wells and their numerical approach, Numer. Math. 83(3) (1999), 325-352.
• [5] Chipot, M. and Collins, C., Numerical approximation in variational problems with potential wells, SIAM J. Numer. Anal. 29(4) (1993), 473-487.
• [6] Chipot, M., Collins, C. and Kinderlehrer, D., Numerical analysis of oscillations in multiple well problems, Numer. Math. 70 (1995), 259-282.
• [7] Chipot, M. and Elfanni, A., On the numerical analysis of some variational problems with nonhomogeneous boundary conditions, Japan J. Indust. Appl. Math. 15 (1998), 345-361.
• [8] Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals, Arch. Rational Mech. Anal. 103 (1988), 237-277.
• [9] Chipot, M. and Müller, S., Sharp energy estimates for finite element approximations of non-convex problems, IUTAM Symposium on Variations of Domain and Free Boundary Problems in Solid Mechanics (P. Argoul, M. Frémond, Q. S. Nguyen, eds), Kluwer, Dordrecht, 1999, 317-325.
• [10] Dacorogna, B., Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989.
• [11] Elfanni, A., Sur quelques questions d’analyse numérique relatives à des problèmes non convexes, Thesis, University of Metz. 1996.
• [12] Friesecke, G., A necessary and sufficient condition for nonattainment and formation of micro structure almost everywhere in scalar variational problems. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 437-471.
• [13] Kohn, R. V. and Müller, S., Branching of twins near an austenite/twinned martensite interface, Philos. Mag. Ser. A 66 (1992), 697-715.
• [14] Raviart, P. A., Thomas, J. M., Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles, Masson, Paris, 1988.