Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Numerical solution of the scalar double-well problem allowing microstructure

Authors: Carsten Carstensen and Petr Plecháč
Journal: Math. Comp. 66 (1997), 997-1026
MSC (1991): Primary 65N15, 65N30, 35J70, 73C60
MathSciNet review: 1415798
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The direct numerical solution of a non-convex variational problem ($P$) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem ($RP$) leading to a (degenerate) convex minimisation problem. The problem ($RP$) has a minimiser $u$ and a related stress field $\sigma = DW^{**}(\nabla {u})$ which is known to coincide with the stress field obtained by solving ($P$) in a generalised sense involving Young measures. If $u_h$ is a finite element solution, $\sigma _h:= D W^{**}(\nabla {u}_h)$ is the related discrete stress field. We prove a priori and a posteriori estimates for $\sigma -\sigma _h $ in $L^{4/3}(\Omega )$ and weaker weighted estimates for $\nabla {u}-\nabla {u}_h$. The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.

References [Enhancements On Off] (What's this?)

  • [Bal89] J.M. Ball, A version of the fundamental theorem for Young measures, In M Rascle, D. Serre, and M. Slemrod, editors, Partial differential equations and continuum models of phase transitions, LNP 344, pages 207-215, 1989. MR 91c:49021
  • [BC94] B. Brighi and M. Chipot, Approximated convex envelope of a function, SIAM J. Numer. Anal., 31, 1994. MR 94m:49049
  • [BJ87] J. M. Ball and R. D. James, Fine phase mixtures as minimisers of energy, Arch. Rational Mech. Anal., 100:13-52, 1987. MR 89c:80005
  • [BJ92] J. M. Ball and R. D. James, Proposed experimental tests of the theory of fine microstructure and the two-well problem, Phil. Trans. R. Soc. Lond. A., 338:389-450, 1992.
  • [BP90] P. Bauman and D. Phillips, A nonconvex variational problem related to change of phase, Appl. Math. Optimization, 21:113-138, 1990. MR 90i:49004
  • [BS94] S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics, Springer Verlag, New York, 1994. MR 95f:65001
  • [CC92] M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells, SIAM J. Numer. Anal., 29:1002-1019, 1992. MR 93j:65171
  • [Chi91] M. Chipot, Numerical analysis of oscillations in nonconvex problems, Numer. Math., 59:747-767, 1991. MR 92h:65164
  • [CL91] C. Collins and M. Luskin, Optimal order error estimates for the finite element approximation of a solution of a nonconvex variational problem, Math. Comp., 57:621-637, 1991. MR 92a:65276
  • [Cle75] P. Clement, Approximation by finite element functions using local regularization, RAIRO Sér. Rouge Anal. Numér., R-2:77-84, 1975. MR 53:4569
  • [Dac89] B. Dacorogna, Direct methods in the calculus of variations, Springer, Berlin, 1989. MR 90e:49001
  • [DK91] P. Pedregal D. Kinderlehrer, Characterisations of Young measures generated by gradients, Arch. Rat. Mech. Anal., 115:329-365, 1991. MR 92k:49089
  • [EG92] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992. MR 92f:28001
  • [Fre90] D. French, On the convergence of finite element approximations of a relaxed variational problem, SIAM J. Num. Anal., 27:419-436, 1990. MR 91f:65167
  • [Fri94] G. Friesecke, A necessary and sufficient condition for non-attainment and formation of microstructure almost everywhere in scalar variational problems, Proc. R. Soc. Edin, 124A:437-471, 1994. MR 96g:49001
  • [GKR86] J. Goodman, R. V. Kohn, and L. Reyna, Numerical study of a relaxed variational problem from optimal design, Comput. Methods Appl. Mech. Engrg., 57:107-127, 1986. MR 88a:73054
  • [GT81] M. E. Gurtin and R. Temam, On the anti-plane shear problem in elasticity, J. Elasticity, 11:197-206, 1981. MR 82d:73030
  • [KP91] D. Kinderlehrer and P. Pedregal, Weak convergence of integrands and the Young measure representation, SIAM J Math. Anal., 23:1-19, 1991. MR 92m:49076
  • [Nas84] S. Nash, Newton-type minimisation via the Lanczos method, SIAM J. Num. Anal., 21:770-787, 1984. MR 86h:65092
  • [NW92] R. A. Nicolaides and N. J. Walkington, Computation of microstructure utilizing Young measure representations, In C.A. Rogers and R.A. Rogers, editors, Recent Advances in Adaptive and Sensory Materials and their Applications, pages 131-141, Lancaster, 1992. Technomic Publishing Co.
  • [NW95] R. A. Nicolaides and N. J. Walkington, Strong convergence of numerical solutions to degenerate variational problems, Math. Comp., 64:117-127, 1995. MR 95m:65183
  • [Ped92] P. Pedregal, Jensen's inequality in calculus of variations, Differential Integral Equations, 7:57-72, 1994. MR 94i:49032
  • [Rou] T. Roubí\v{c}ek, Relaxation in optimization theory and variational calculus, DeGruyter, Berlin 1997.
  • [Ver94] R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretization of elliptic equations, Math. Comp., 62:445-475, 1994. MR 94j:65136

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N15, 65N30, 35J70, 73C60

Retrieve articles in all journals with MSC (1991): 65N15, 65N30, 35J70, 73C60

Additional Information

Carsten Carstensen
Affiliation: Mathematical Institute, Oxford University, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom

Petr Plecháč
Affiliation: Mathematical Institute, Oxford University, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom

Keywords: Non-convex minimisation, Young measures, microstructure
Received by editor(s): May 8, 1995
Received by editor(s) in revised form: May 3, 1996
Additional Notes: The work of the first author was supported by the EC under HCM ERB CH BG CT 920007, the work of the second author was supported under EPSRC grant GR/JO3466.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society