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A numerical method for detecting singular minimizers

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Summary

A numerical method for computing minimizers in one-dimensional problems of the calculus of variations is described. Such minimizers may have unbounded derivatives, even when the integrand is smooth and regular. In such cases, because of the Lavrentiev phenomenon, standard finite element methods may fail to converge to a minimizer. The scheme proposed is shown to converge to an absolute minimizer and is tested on an example. The effect of quadrature is analyzed. The implications for higher-dimensional problems, and in particular for fracture in nonlinear elasticity, are discussed.

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, EH144AS, Edinburgh, Scotland

    J. M. Ball

  2. Department of Electrical Engineering, Imperial College of Science and Technology, Queen's Gate, SW72BZ, London, UK

    G. Knowles

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  1. J. M. Ball
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  2. G. Knowles
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Ball, J.M., Knowles, G. A numerical method for detecting singular minimizers. Numer. Math. 51, 181–197 (1987). https://doi.org/10.1007/BF01396748

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