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A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge–Ampère Equation

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Abstract

In the present article we extend to the three-dimensional elliptic Monge–Ampère equation the method discussed in Glowinski et al. (J Sci Comput 79:1–47, 2019) for the numerical solution of its two-dimensional variant. As in Glowinski et al. (2019) we take advantage of an equivalent divergence formulation of the Monge–Ampère equation, involving the cofactor matrix of the Hessian of the solution. We associate with the above divergence formulation an initial value problem, well suited to time discretization by operator splitting and space approximation by low order mixed finite element methods. An important ingredient of our methodology is forcing the positive semi-definiteness of the approximate Hessian by a hard thresholding eigenvalue projection. The resulting method is robust and easy to implement. It can handle problems with smooth and non-smooth solutions on domains with curved boundary. Using piecewise affine approximations for the solution and its six second-order derivatives, one can achieve second-order convergence rates for problems with smooth solutions.

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Acknowledgements

The work of R. Glowinski was partially supported by the Hong Kong Kennedy Wong foundation. The work of S. Leung was partially supported by the Hong Kong RGC Grants 16302819 and 16309316. The work of J. Qian was partially supported by NSF.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30332, USA

    Hao Liu

  2. Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX, 77204, USA

    Roland Glowinski

  3. Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Hong Kong, People’s Republic of China

    Roland Glowinski

  4. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China

    Shingyu Leung

  5. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian

Authors
  1. Hao Liu
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  2. Roland Glowinski
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  3. Shingyu Leung
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  4. Jianliang Qian
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Correspondence to Hao Liu.

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Liu, H., Glowinski, R., Leung, S. et al. A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge–Ampère Equation. J Sci Comput 81, 2271–2302 (2019). https://doi.org/10.1007/s10915-019-01080-4

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  • DOI: https://doi.org/10.1007/s10915-019-01080-4

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