Solving the 2-D elliptic Monge-Ampere equation by a Kansa's method
Solving the 2-D elliptic Monge-Ampere equation by a Kansa's method
复制标题
用 Kansa 法求解二维椭圆 Monge-Ampere 方程
DOI：
10.1007/s10255-017-0656-3
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发表时间：
2017
影响因子：
0.8
通讯作者：
Liu Zhiyong
中科院分区：
文献类型：
--
作者：
Li Qin;Liu Zhiyong
本文设计了一种Kansa方法来数值求解Monge-Ampère方程。原始Kansa方法是一种无网格方法，它利用一些径向基函数的组合(如Hardy的MQ)来逼近线性抛物型、双曲型和椭圆型问题的解。但当用于求解非线性偏微分方程组时，这种方法就变差了。我们用一些三次多项式的组合来逼近局部三角子域上的解。然后，给定的问题可以在每个子域中独立计算。证明了新方法解椭圆型Monge-Ampère方程的稳定性和收敛性。最后，通过数值实验对理论结果进行了验证。
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.