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Quadratic Finite Element Approximations of the Monge-Ampère Equation

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Abstract

We prove several new results of the C 0 finite element method introduced in (S.C. Brenner et al., Math. Comput. 80:1979–1995, 2011) for the fully nonlinear Monge-Ampère equation. These include the convergence of quadratic finite element approximations, W 2,p quasi-optimal error estimates, localized pointwise error estimates, and convergence of Newton’s method with explicit dependence on the discretization parameter. Numerical experiments are presented which back up the theoretical results.

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Acknowledgements

The work of this author was supported by the National Science Foundation under grant number DMS-1115421.

The author would like to thank Susanne C. Brenner and Li-Yeng Sung for helpful discussions.

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  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Michael Neilan

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Neilan, M. Quadratic Finite Element Approximations of the Monge-Ampère Equation. J Sci Comput 54, 200–226 (2013). https://doi.org/10.1007/s10915-012-9617-4

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  • DOI: https://doi.org/10.1007/s10915-012-9617-4

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