## Numerical solution of the scalar double-well problem allowing microstructure

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- by Carsten Carstensen and Petr Plecháč PDF
- Math. Comp.
**66**(1997), 997-1026 Request permission

## 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

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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
- 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.
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp.
**66**(1997), 997-1026 - MSC (1991): Primary 65N15, 65N30, 35J70, 73C60
- DOI: https://doi.org/10.1090/S0025-5718-97-00849-1
- MathSciNet review: 1415798