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Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

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Abstract

We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Ampére equation. Mainly we use the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author upon request.

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Acknowledgements

The authors would like to thanks anonymous referees for their valuable comments. In addition, the authors would like to thank Dr. Gerard Awanou for generosity for providing references [22] and [55].

Funding

Dr. Ming-Jun Lai is supported by the Simons Foundation collaboration grant #864439.

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  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, Georgia

    Ming-Jun Lai & Jinsil Lee

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  1. Ming-Jun Lai
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Lai, MJ., Lee, J. Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation. J Sci Comput 95, 56 (2023). https://doi.org/10.1007/s10915-023-02183-9

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