On the numerical analysis of some variational problems with nonhomogeneous boundary conditions

Springer Science and Business Media LLC - Tập 15 - Trang 345-361 - 1998
M. Chipot1, A. Elfanni2
1Institut für Angewandte Mathematik, Universität Zürich, Zürich, Switzerland
2Centre d’Analyse Non Linéaire, Université de Metz, Metz Cedex 01, France

Tóm tắt

The goal of this note is to expose a new techniques to get energy estimates for nonconvex problems with nonlinear boundary conditions in term of the mesh size of a Lagrange finite elements method.

Tài liệu tham khảo

J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal.,100 (1987), 13–52. J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructures. Phil. Trans. Roy. Soc. London A,338 (1992), 350–389. J.M. Ball and F. Murat,W 1,p quasiconvexity and variational problems for multiples integrals. J. Funt. Anal.,58 (1984), 225–253. B. Brighi, Sur quelques problèmes de calcul des variations et l’approximation de leur fonctionnelles relaxées. Thesis, University of Metz, 1991. B. Brighi and M. Chipot, Approximation in nonconvex problems. Proceedings of the first European Conference on Elliptic and Parabolic Problems, Pont-à-Mousson, June 1991. Pitman research notes in mathematics, #267, 1992, 150–157. B. Brighi and M. Chipot, Approximated convex envelope of a function. SIAM J. Numer. Anal.,31, 1 (1994), 128–148. M. Chipot, Hyperelasticity for crystals. Eur. J. Appl. Math.,1 (1990), 113–129. M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numerische Mathematik,59 (1991), 747–767. M. Chipot, Energy estimates for variational problems with nonhomogeneous boundary conditions. Nonlinear mathematical problems in industry, Gakuto International Series, Mathematical Sciences and Applications,2, 1993, 473–487. M. Chipot and C. Collins, Numerical approximation in variational problems with potential wells. SIAM J. Numer. Anal.,29, 4 (1993), 473–487. M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numerische Mathematik,70 (1995), 259–282. M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal.,103 (1988), 237–277. M. Chipot and W. Li, Variational problems with potential wells and nonhomogeneous boundary conditions. Calculus of variations, homogenization and continuum mechanics (eds. G. Bouchitté, G. Buttazzo and P. Suquet), Series on Advances in Mathematics for Applied Sciences,18, World Scientific, 1994, 149–168. M. Chipot and S. Müller, Sharp energy estimates to finite element approximations for nonconvex problems (to appear). C. Collins, Thesis, University of Minnesota, 1990. C. Collins, D. Kinderlehrer and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potentiel. SIAM J. Numer. Anal.,28 (1991), 321–332. C. Collins and M. Luskin, The computation of austenetic-martensitic phase transition. Partial Differential Equations and Continum Models of Phase Transitions (eds. M. Rascle, D. Serre and M. Slemrod), Lecture Notes in Physics, #344, Springer-Verlag, Berlin, New York, 1989, 34–50. C. Collins and M. Luskin, Computational results for phase transitions in shape memory materials. Smart Materials, Structure, and Mathematical Issues (ed. C. Rogers), Technomic Publishing Co., Lancaster, Pennsylvania, 1989, 198–215. C. Collins and M. Luskin, Numerical modeling of the microstructure of crystals with symmetry-related variants. Proceedings of the ARO US-Japan Workshop on Smart/Intelligent Materials and Systems, Technomic Publishing Co., Lancaster, Pennsylvania, 1990, 309–318. C. Collins and M. Luskin, Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp.,57 (1991), 621–637. B. Dacorogna, Weak continuity and weak lower semicontinuity of non linear functionals. Springer Lectures Notes, #922, 1982. A. Elfanni, Sur quelques questions d’analyse numérique relatives à des problèmes non convexes. Thesis, University of Metz, 1996. J.L. Ericksen, Some constrainted elastic crystals. Material Instabilities in Continum Mechanics and Related Problems (ed. J.M. Ball), Oxford University Press, Oxford, 1987, 119–137. I. Fonseca, Variational methods for elastic crystals. Arch. Rational Mech. Anal.,97 (1985), 189–220. I. Fonseca, The lower quasiconvex envelope of stored energy function for an elastic crystal. J. Math. Pures et Appl.,67 (1988), 175–195. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer Verlag, 1985. R.D. James, Basic principles for the improvement of shape-memory and related materials. Smart Materials, Structure, and Mathematical Issues (ed. C. Rogers), Technomic Publishing Co., Lancaster, Pennsylvania, PA, 1989. R.D. James, Microstructure and weak convergence. Material Instabilities in Continum Mechanics and Related Problems (ed. J.M. Ball), Oxford University Press, Oxford, 1987, 175–196. R.D. James and D. Kinderlehrer, Theory of diffusionless phase transitions. Partial Differential Equations and Continum Models of Phase Transitions (eds. M. Rascle, D. Serre and M. Slemrod), Lecture Notes in Physics, #344, Springer-Verlag, Berlin, New York, 1989, 51–84. D. Kinderlehrer, Remarks about equilibrium configurations of crystals. Material Instabilities in Continum Mechanics and Related Problems (ed. J.M. Ball), Oxford University Press, Oxford, 1987, 217–242. R. Kohn, The relationship between linear and nonlinear variational models of coherent phase transitions. Proceedings of seventh Army Conference on Applied Mathematics and Computing, West Point, June 1989. W. Li, Thesis, University of Metz, 1993. P.A. Raviart and J.M. Thomas, Introduction à l’analyse numérique des équations aux dérivées partielles. Masson, Paris, 1988.