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S. Atluri, S. Shen (2002)
The meshless local Petrov-Galerkin (MLPG) method
(2008)
The meshless local Petrov - Galerkin ( MLPG ) method for the generalized two - dimensional nonlinear Schrödinger equation ”
(2002)
The meshless local Petrov-Galerkin method: a simple & less-costly
Y. Lacasse, F. Maltais (2005)
From the authorsEuropean Respiratory Journal, 26
(2009)
A local boundary integral equation
(2011)
A meshless local boundary integral
S. Atluri, S. Shen (2002)
The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element MethodsCmes-computer Modeling in Engineering & Sciences, 3
W. Nicomedes, R. Mesquita, F. Moreira (2012)
The Meshless Local Petrov–Galerkin Method in Two-Dimensional Electromagnetic Wave AnalysisIEEE Transactions on Antennas and Propagation, 60
W. Nicomedes, R. Mesquita, F. Moreira (2010)
A Meshless Local Boundary Integral Equation method for three dimensional scalar problemsDigests of the 2010 14th Biennial IEEE Conference on Electromagnetic Field Computation
Guirong Liu (2002)
Mesh Free Methods: Moving Beyond the Finite Element Method
W. Nicomedes, R. Mesquita, F. Moreira (2009)
A Local Boundary Integral Equation (LBIE) Method in 2D electromagnetic wave scattering, and a meshless discretization approach2009 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference (IMOC)
(2002)
The meshless local Petrov - Galerkin method : a simple & less - costly alternative to the finite - element and boundary element methods ”
M. Dehghan, D. Mirzaei (2008)
The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equationEngineering Analysis With Boundary Elements, 32
W. Nicomedes, R. Mesquita, F. Moreira (2011)
A Meshless Local Petrov–Galerkin Method for Three-Dimensional Scalar ProblemsIEEE Transactions on Magnetics, 47
S. Jun (2004)
Meshfree implementation for the real‐space electronic‐structure calculation of crystalline solidsInternational Journal for Numerical Methods in Engineering, 59
Purpose – The purpose of this paper is to solve both eigenvalue and boundary value problems coming from the field of quantum mechanics through the application of meshless methods, particularly the one known as meshless local Petrov‐Galerkin (MLPG). Design/methodology/approach – Regarding eigenvalue problems, the authors show how to apply MLPG to the time‐independent Schrödinger equation stated in three dimensions. Through a special procedure, the numerical integration of weak forms is carried out only for internal nodes. The boundary conditions are enforced through a collocation method. The final result is a generalized eigenvalue problem, which is readily solved for the energy levels. An example of boundary value problem is described by the time‐dependent nonlinear Schrödinger equation. The weak forms are again stated only for internal nodes, whereas the same collocation scheme is employed to enforce the boundary conditions. The nonlinearity is dealt with by a predictor‐corrector scheme. Findings – Results show that the combination of MLPG and a collocation scheme works very well. The numerical results are compared to those provided by analytical solutions, exhibiting good agreement. Originality/value – The flexibility of MLPG is made explicit. There are different ways to obtain the weak forms, and the boundary conditions can be enforced through a number of ways, the collocation scheme being just one of them. The shape functions used to approximate the solution can incorporate modifications that reflect some feature of the problem, like periodic boundary conditions. The value of this work resides in the fact that problems from other areas of electromagnetism can be attacked by the very same ideas herein described.
COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering – Emerald Publishing
Published: Nov 15, 2011
Keywords: Meshless methods; Quantum mechanics; Schrödinger equation; Eigenvalues and Eigenfunctions; MLPG; Numerical analysis
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