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

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

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]**R. A. Adams,*Sobolev spaces*, Academic Press, New York, 1975. MR**0450957 (56:9247)****[2]**J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart,*On the dynamics of fine structure*, preprint, 1990. MR**1102830 (92d:35194)****[3]**J. M. Ball and R. D. James,*Fine phase mixtures as minimizers of energy*, Arch. Rational Mech. Anal.**100**(1987), 13-52. MR**906132 (89c:80005)****[4]**-,*Experimental tests of a theory of fine microstructure*, preprint, 1990.**[5]**M. Chipot and D. Kinderlehrer,*Equilibrium configurations of crystals*, Arch. Rational Mech. Anal.**103**(1988), 237-277. MR**955934 (90a:73037)****[6]**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. MR**1087507 (92c:73015)****[7]**C. Collins and M. Luskin,*The computation of the austenitic-martensitic phase transition*, Partial Differential Equations and Continuum Models of Phase Transitions (M. Rascle, D. Serre, and M. Slemrod, eds.), Lecture Notes in Phys., vol. 344, Springer-Verlag, Berlin and New York, 1989, pp. 34-50. MR**1036062 (90k:80007)****[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*, Holt, Rinehart, and Winston, New York, 1965. MR**0221256 (36:4308)****[11]**J. L. Ericksen,*Some constrained elastic crystals*, Material Instabilities in Continuum Mechanics and Related Problems (J. M. Ball, ed.), Oxford Univ. Press, Oxford, 1987, pp. 119-137. MR**970522 (90a:73145)****[12]**-,*Constitutive theory for some constrained elastic crystals*, Internat. J. Solids and Structures**22**(1986), 951-964.**[13]**I. Fonseca,*Variational methods for elastic crystals*, Arch. Rational Mech. Anal.**97**(1985), 189-220. MR**862547 (87i:73029)****[14]**-,*The lower quasiconvex envelope of the stored energy function for an elastic crystal*, J. Math. Pures Appl.**67**(1988), 175-195. MR**949107 (89g:73042)****[15]**D. A. French,*On the convergence of finite element approximations of a relaxed variational problem*, SIAM J. Numer. Anal.**27**(1990), 419-436. MR**1043613 (91f:65167)****[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 and Related Problems (J. M. Ball, ed.), Oxford Univ. Press, Oxford, 1987, pp. 175-196. MR**970525 (90k:73014)****[18]**R. D. James and D. Kinderlehrer,*Theory of diffusionless phase transitions*, Partial Differential Equations and Continuum Models of Phase Transitions (M. Rascle, D. Serre, and M. Slemrod, eds.), Lecture Notes in Phys., vol. 344,, Springer-Verlag, Berlin, and New York, 1989, pp. 51-84. MR**1036063 (91e:73011)****[19]**D. Kinderlehrer,*Remarks about equilibrium configurations of crystals*, Material Instabilities in Continuum Mechanics and Related Problems (J. M. Ball, ed.), Oxford Univ. Press, Oxford, 1987, pp. 217-242. MR**970527****[20]**R. Kohn,*The relationship between linear and nonlinear variational models of coherent phase transitions*, Proceedings of the Seventh Army Conference on Applied Mathematics and Computing (F. Dressel, ed.), West Point, 1989. MR**1057840****[21]**W. Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill, New York, 1987. MR**924157 (88k:00002)****[22]**L. Tartar,*Compensated compactness and applications to partial differential equations*, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium IV, Research Notes 39 (R. J. Knops, ed.), Pitman, London, 1978, pp. 136-212. MR**584398 (81m:35014)**

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