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Convergence of adaptive finite element methods for a nonconvex double-well minimization problem


Authors: Carsten Carstensen and Georg Dolzmann
Journal: Math. Comp. 84 (2015), 2111-2135
MSC (2010): Primary 65N12, 65N30
DOI: https://doi.org/10.1090/S0025-5718-2015-02947-0
Published electronically: February 26, 2015
MathSciNet review: 3356021
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Abstract: This paper focuses on the numerical analysis of a nonconvex variational problem which is related to the relaxation of the two-well problem in the analysis of solid-solid phase transitions with incompatible wells and dependence on the linear strain in two dimensions. The proposed approach is based on the search for minimizers for this functional in finite element spaces with Courant elements and with successive loops of the form SOLVE, ESTIMATE, MARK, and REFINE. Convergence of the total energy of the approximating deformations and strong convergence of all except one component of the corresponding deformation gradients is established. The proof relies on the decomposition of the energy density into a convex part and a null-Lagrangian. The key ingredient is the fact that the convex part satisfies a convexity property which is stronger than degenerate convexity and weaker than uniform convexity. Moreover, an estimator reduction property for the stresses associated to the convex part in the energy is established.


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Additional Information

Carsten Carstensen
Affiliation: Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany — and — Department of Computational Science and Engineering, Yonsei University, Seoul, Korea

Georg Dolzmann
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

DOI: https://doi.org/10.1090/S0025-5718-2015-02947-0
Received by editor(s): December 21, 2012
Received by editor(s) in revised form: September 21, 2013
Published electronically: February 26, 2015
Additional Notes: This work was partially supported by the Deutsche Forschungsgemeinschaft through the Forschergruppe 797 “Analysis and computation of microstructure in finite plasticity”, projects CA 151/19 (first author) and DO 633/2 (second author). The work was finalized while the first author enjoyed the kind hospitality of the Oxford Centre for Nonlinear PDE.
Article copyright: © Copyright 2015 American Mathematical Society