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Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method

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Abstract

In this paper, a Kansa’s method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa’s method is a meshfree method which applying the combination of some radial basis functions (such as Hardy’s MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.

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Author information

Authors and Affiliations

  1. School of Science, Beijing Technology and Business University, Beijing, 100084, China

    Qin Li

  2. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Zhi-yong Liu

Authors
  1. Qin Li
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  2. Zhi-yong Liu
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Correspondence to Qin Li.

Additional information

The first author is supported in part by the National Natural Science Foundations of China (No.11426039, 11571023, 11471329). The second author is partially supported by the National Natural Science Foundation of China (No.11501313), the Natural Science Foundation of Ningxia Province (No.NZ15005), and the Science Research Project of Ningxia Higher Education (No.NGY2016059).

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Li, Q., Liu, Zy. Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method. Acta Math. Appl. Sin. Engl. Ser. 33, 269–276 (2017). https://doi.org/10.1007/s10255-017-0656-3

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  • DOI: https://doi.org/10.1007/s10255-017-0656-3

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