Abstract

Non-quasiconvex variational problems require, in general, a relaxation to ensure existence of their solutions. Here a relaxation by (generalized) Young functionals is used and then a finite-element method for a direct numerical approximation of the relaxed problems is developed. Beside a mere convergence, also some error estimates are analysed and a comparison with other methods is discussed.

Author information

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T. Roubíček

Mathematical Institute, Charles University, Sokolovská 83, CZ-186 00 Praha 8, and Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic.

Keywords

  • Vectorial variational problem
  • relaxation
  • generalized Young functionals
  • finite-element approximation

References

  1. J.M. Ball: On the calculus of variations and sequentially weakly continuous maps. In: Proc. Conf. Ordinary and Partial Differential Equations. (W.N.Everitt, B.D.Sleeman, eds.), Lecture Notes in Math. 564, Springer, Berlin (1976) 13–25.https://doi.org/10.1007/BFb0087322Google Scholar
  2. J.M. Ball, R.D. James: Fine phase mixtures as minimizers of energy, Archive Rat. Mech. Anal. 100 (1988) 13–52.https://doi.org/10.1007/BF00281246Google Scholar
  3. J.M. Ball, R.D. James: Proposed experimental tests of a theory of fine microstructure and the two-well problem, Phil. Trans. Royal Soc. London A 338 (1992) 389–450.https://doi.org/10.1098/rsta.1992.0013Google Scholar
  4. B. Brighi, M. Chipot: Approximation in nonconvex problems. In: “Progress in PDEs: Calculus of Variations, Applications” (Eds. C.Bandle et al.), Pitman Res. Notes in Math. Sci. 267 (1992) Longmann, 150–157.Google Scholar
  5. B. Brighi, M. Chipot: Approximated convex envelope of a function. A preprint.Google Scholar
  6. M. Chipot: Numerical analysis of oscillations in conconvex problems, Numer. Math. 59 (1991) 747–767.https://doi.org/10.1007/BF01385808Google Scholar
  7. M. Chipot, C. Collins: Numerical approximations in variational problems with potential wells, SIAM J. Numer. Anal. 29 (1992) 1002–1019.https://doi.org/10.1137/0729061Google Scholar
  8. M. Chipot, C. Collins and D. Kinderlehrer: Numerical analysis of oscillations in multiple well problems, Numer. Math. 70 (1995) 259–282.https://doi.org/10.1007/s002110050119Google Scholar
  9. M. Chipot, D. Kinderlehrer: Equilibrium configurations of crystals, Arch. Rational Mech. Anal. 103 (1988) 237–277.https://doi.org/10.1007/BF00251759Google Scholar
  10. P.G. Ciarlet: Basic error estimates for elliptic problems, In: Handbook of Numerical Analysis, Vol. II, Finite element methods (Part 1), P.G. Ciarlet, J.L. Lions, eds., North-Holland, Amsterdam, 1991.https://doi.org/10.1016/S1570-8659(05)80039-0Google Scholar
  11. C. Collins, D. Kinderlehrer and M. Luskin: Numerical approximation of the solution of a variational problem with a double well potential, SIAM J. Numer. Anal. 28 (1991) 321–332.https://doi.org/10.1137/0728018Google Scholar
  12. C. Collins, M. Luskin: Optimal order error estimates for finite element approximation of the solution of a nonconvex variational problem, Math. Comp. 57 (1991) 621–637.https://doi.org/10.1090/S0025-5718-1991-1094944-0Google Scholar
  13. C. Collins, M. Luskin: Numerical modelling of the microstructure of crystals with symmetryrelated variants. In: US-Japan Workshop on Smart/Inteligent Materials and Systems (Eds. I.Ahmad et al.), Technomic Publ. Comp., Lancaster (1990) 309–318.Google Scholar
  14. C. Collins, M. Luskin and J. Riordan: Computational results for a two-dimensional model of crystalline microstructure. In: IMA Vol. in Math. and Applications “Microstructure and Phase Transitions” (Eds. R.James, D.Kinderlehrer, M.Luskin), Springer, New York (1993) 51–56.https://doi.org/10.1007/978-1-4613-8360-4_4Google Scholar
  15. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin, 1989.https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  16. M. Kružík: Numerical approach to double-well problems (submitted).Google Scholar
  17. M. Luskin: Numerical analysis of microstructure for crystals with a nonconvex energy density. In: The Metz Days Surveys 1989-90 (M.Chipot, J. Saint Jean Paulin, eds.), Pitman Res. Notes in Math., Longman, Harlow (1991) 156–165.Google Scholar
  18. S. Müller: Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations. In: Lecture Notes in Physics 359 (K. Kirchgäsner, ed.), Springer, Berlin (1988) 31–44.https://doi.org/10.1007/3-540-52595-5_83Google Scholar
  19. C.B. Morrey, Jr.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25–53.https://doi.org/10.2140/pjm.1952.2.25Google Scholar
  20. R.A. Nicolaides, N.J. Walkington: Computation of microstructure utilizing Young measure representations. In: Recent Advances in Adaptive and Sensory Materials and their Applications (C.A.Rogers, R.A.Roegers, eds.), Technomic Publ. (1992) 131–141.Google Scholar
  21. R.A. Nicolaides, N.J. Walkington: Strong convergence of numerical solutions to degenerate variational problems, Res. Report 92-NA-027, Center for Nonlinear Analysis, Carnegie Mellon University, 1992.Google Scholar
  22. Y.G. Reshetnyak: On the stability of conformal mappings in multidimensional spaces. Siberian Math. J. 8 (1967) 69–85.https://doi.org/10.1007/BF01040573Google Scholar
  23. T. Roubíček: Finite element approximation of a microstructure evolution, Math. Methods in the Appl. Sci. 17 (1994) 377–393.https://doi.org/10.1002/mma.1670170505Google Scholar
  24. T. Roubíček: Effective characterization of generalized Young measures generated by gradients. Bollettino Uni. Mat. Italiana 9-B (1995) 755–779.Google Scholar
  25. T. Roubíček: Relaxation of vectorial variational problems, Matematica Bohemica 120 (1995) 411–430.https://doi.org/10.21136/MB.1995.126087Google Scholar
  26. T. Roubíček: Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin, 1997.https://doi.org/10.1515/9783110811919Google Scholar
  27. T. Roubíček, K.-H. Hoffmann: Theory of convex local compactifications with applications to Lebesgue spaces, Nonlinear Analysis, Th., Methods, Appl. 20 (1995) 607–628.https://doi.org/10.1016/0362-546X(94)00150-GGoogle Scholar
  28. L.C. Young: Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Classe III 30 (1937) 212–234.Google Scholar
  29. J. Zowe, S. Kurcyusz: Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim. 5 (1979) 49–62.https://doi.org/10.1007/BF01442543Google Scholar

Suggested citation

T. Roubíček. “Numerical Approximation of Relaxed Variational Problems.” Journal of Convex Analysis 3 (1996), No. 2, 329–347. https://doi.org/10.68381/jca03022

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Published by Heldermann Verlag, 1996. Rights now held by Banach Press.