Optimal-order error estimates for the finite element approximation of the solution of a nonconvex variational problem

Authors:
Charles Collins and Mitchell Luskin

Journal:
Math. Comp. **57** (1991), 621-637

MSC:
Primary 65N15; Secondary 35J20, 35J70, 65N30, 73C99, 73V05

MathSciNet review:
1094944

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials. The solution of these variational problems must be described in terms of a microstructure rather than in terms of a deformation. Moreover, the numerical approximation of the deformation gradient often does not converge strongly as the mesh is refined. Nevertheless, the probability distribution of the deformation gradients near each material point does converge. Recently we introduced a metric to analyze this convergence. In this paper, we give an optimal-order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart,*On the dynamics of fine structure*, J. Nonlinear Sci.**1**(1991), no. 1, 17–70. MR**1102830**, 10.1007/BF01209147**[3]**J. M. Ball and R. D. James,*Fine phase mixtures as minimizers of energy*, Arch. Rational Mech. Anal.**100**(1987), no. 1, 13–52. MR**906132**, 10.1007/BF00281246**[4]**-,*Experimental tests of a theory of fine microstructure*, preprint, 1990.**[5]**Michel Chipot and David Kinderlehrer,*Equilibrium configurations of crystals*, Arch. Rational Mech. Anal.**103**(1988), no. 3, 237–277. MR**955934**, 10.1007/BF00251759**[6]**Charles Collins, David Kinderlehrer, and Mitchell Luskin,*Numerical approximation of the solution of a variational problem with a double well potential*, SIAM J. Numer. Anal.**28**(1991), no. 2, 321–332. MR**1087507**, 10.1137/0728018**[7]**Charles Collins and Mitchell Luskin,*The computation of the austenitic-martensitic phase transition*, PDEs and continuum models of phase transitions (Nice, 1988) Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 34–50. MR**1036062**, 10.1007/BFb0024934**[8]**-,*Computational results for phase transitions in shape memory materials*, Smart Materials, Structures, and Mathematical Issues (C. Rogers, ed.), Technomic Publishing Co., Lancaster, PA, 1989, pp. 198-215.**[9]**-,*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, PA, 1990, pp. 309-318.**[10]**R. E. Edwards,*Functional analysis. Theory and applications*, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR**0221256****[11]**J. L. Ericksen,*Some constrained elastic crystals*, Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 119–135. MR**970522****[12]**-,*Constitutive theory for some constrained elastic crystals*, Internat. J. Solids and Structures**22**(1986), 951-964.**[13]**Irene Fonseca,*Variational methods for elastic crystals*, Arch. Rational Mech. Anal.**97**(1987), no. 3, 189–220. MR**862547**, 10.1007/BF00250808**[14]**Irene Fonseca,*The lower quasiconvex envelope of the stored energy function for an elastic crystal*, J. Math. Pures Appl. (9)**67**(1988), no. 2, 175–195. MR**949107****[15]**Donald A. French,*On the convergence of finite-element approximations of a relaxed variational problem*, SIAM J. Numer. Anal.**27**(1990), no. 2, 419–436. MR**1043613**, 10.1137/0727025**[16]**R. James,*Basic principles for the improvement of shape-memory and related materials*, Smart Materials, Structures, and Mathematical Issues (C. Rogers, ed.), Technomic Publishing Co., Lancaster, PA, 1989.**[17]**R. D. James,*Microstructure and weak convergence*, Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 175–196. MR**970525****[18]**Richard James and David Kinderlehrer,*Theory of diffusionless phase transitions*, PDEs and continuum models of phase transitions (Nice, 1988) Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 51–84. MR**1036063**, 10.1007/BFb0024935**[19]**David Kinderlehrer,*Remarks about equilibrium configurations of crystals*, Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 217–241. MR**970527****[20]**Robert V. Kohn,*The relationship between linear and nonlinear variational models of coherent phase transitions*, Transactions of the Seventh Army Conference on Applied Mathematics and Computing (West Point, NY, 1989) ARO Rep., vol. 90, U.S. Army Res. Office, Research Triangle Park, NC, 1990, pp. 279–304. MR**1057840****[21]**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****[22]**L. Tartar,*Compensated compactness and applications to partial differential equations*, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR**584398**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N15,
35J20,
35J70,
65N30,
73C99,
73V05

Retrieve articles in all journals with MSC: 65N15, 35J20, 35J70, 65N30, 73C99, 73V05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1991-1094944-0

Keywords:
Finite element method,
error estimates,
numerical approximation,
variational problem,
nonconvex

Article copyright:
© Copyright 1991
American Mathematical Society