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A meshfree method for solving the Monge–Ampère equation

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Abstract

This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and collocation in the domain and on the boundary. Convergence rates may be up to exponential, depending on the smoothness of the true solution, and this is demonstrated numerically and proven theoretically, applying a sufficiently fine collocation discretization. A much more thorough investigation of meshless methods for fully nonlinear problems is in preparation.

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Acknowledgements

The authors thank the referees for several suggestions improving the paper, in particular the presentation of the numerical results.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Universität Marburg, Arbeitsgruppe Numerik, Hans Meerwein Straße, 35032, Lahnberge, Marburg, Germany

    Klaus Böhmer

  2. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16-18, 37073, Göttingen, Germany

    Robert Schaback

Authors
  1. Klaus Böhmer
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  2. Robert Schaback
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Correspondence to Robert Schaback.

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Böhmer, K., Schaback, R. A meshfree method for solving the Monge–Ampère equation. Numer Algor 82, 539–551 (2019). https://doi.org/10.1007/s11075-018-0612-1

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  • DOI: https://doi.org/10.1007/s11075-018-0612-1

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