Solving second order non-linear elliptic partial differential equations using generalized finite difference method

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• 作者：L. Gavete^a ; ^{lu.gavete@upm.es} ; F. Ureñ ; a^b ; ^{fuprieto@terra.com} ; J.J. Benito^b ; ^{jbenito@ind.uned.es} ; A. Garcí ; a^a ; ^{angelochuri@gmail.com} ; M. Ureñ ; a^b ; ^{miguelurenya@gmail.com} ; E. Salete^b ; ^{esalete@ind.uned.es}
• 关键词：Meshless methods ; Generalized finite difference method ; Non-linear elliptic partial differential equations ; Newton&ndash ; Raphson method
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：July 2017
• 年：2017
• 卷：318
• 期：Complete
• 页码：378-387
• 全文大小：1300 K
• 卷排序：318

文摘

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (pde’s): wave propagation, advection–diffusion, plates, beams, etc.The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear elliptic pde’s.This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the pde’s are based on the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Also it is shown that in 2D a regular neighbourhood of eight nodes can be regarded as a generalization of a classical finite difference formula with a sixth order truncation error.This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.