Solving Monge-Ampère equation in 2D and 3D by Generalized Finite Difference Method

In this paper we derive the discretization of the nonlinear Monge-Ampère equation by means of the explicit formulae of the meshless Generalized Finite Difference Method (GFDM) in both two and three dimensional settings (2D and 3D). To do so we implement the Cascadic iterative algorithm. We provide a...

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Bibliographic Details
Published in:Engineering analysis with boundary elements 2021-03, Vol.124, p.52-63
Main Authors: Benito, J.J., García, A., Gavete, L., Negreanu, M., Ureña, F., Vargas, A.M.
Format: Article
Language:English
Subjects:
Generalized Finite Difference Method
Iterative algorithm
Monge-Ampère equation
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Summary:In this paper we derive the discretization of the nonlinear Monge-Ampère equation by means of the explicit formulae of the meshless Generalized Finite Difference Method (GFDM) in both two and three dimensional settings (2D and 3D). To do so we implement the Cascadic iterative algorithm. We provide a rigorous proof of the consistency of the GFDM for this elliptic equation and present several examples in 2D and 3D, where the method shows its potential and accuracy. We provide a discussion on the number of total nodes in the domain and the number of local supporting nodes. Also, we give an example with physical meaning where we find a surface whose Gaussian curvature is given.
ISSN:0955-7997
1873-197X
DOI:10.1016/j.enganabound.2020.12.007