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A meshfree method for solving the Monge–Ampère equation

  • Klaus Böhmer
  • Robert Schaback
Original Paper
  • 23 Downloads

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.

Keywords

Collocation Fully nonlinear PDE Monge–Ampère Nonlinear optimizer MATLAB implementation Convergence Error analysis Error estimates 

Mathematics Subject Classification (2010)

35J36 65D99 65N12 65N35 

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Notes

Acknowledgements

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

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikUniversität Marburg, Arbeitsgruppe NumerikLahnbergeGermany
  2. 2.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

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