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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Convergence of adaptive finite element methods for a nonconvex double-well minimization problem
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by Carsten Carstensen and Georg Dolzmann PDF
Math. Comp. 84 (2015), 2111-2135 Request permission

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
  • 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.
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 2111-2135
  • MSC (2010): Primary 65N12, 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-2015-02947-0
  • MathSciNet review: 3356021