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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
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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.


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Additional Information

Carsten Carstensen
Affiliation: Mathematical Institute, Oxford University, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom
Email: cc@numerik.uni-kiel.de

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

DOI: https://doi.org/10.1090/S0025-5718-97-00849-1
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