Solving a scattering problem in near field optics by a least-squares method

Li, Tian; Sun, Yubao

doi:10.1016/j.enganabound.2016.01.006

Cited by 5 publications

(1 citation statement)

References 31 publications

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“…The least squares concept has been widely applied for solving PDEs. For example, Luan and Sun used the least squares method to solve a scattering problem in near eld optics [24]. Recently, Discrete Least Squares Meshless (DLSM) method was proposed by Afshar et al to solve the elliptic and convection-dominated problems [25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

2017

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Abstract. A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as atruly mesh-free method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation is deployed, which needs to calculate the costly second derivatives of the MLS shape functions. In the proposed MDLSM method, the complex and costly second derivatives of shape functions are not required. Furthermore, using the mixed formulation, both unknown parameters and their gradients are simultaneously obtained circumventing the need for post-processing procedure performed in irreducible formulation to calculate the gradients. Therefore, the accuracy of gradients of unknown parameters is increased. In MDLSM method, the set of simultaneous algebraic equations is built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is de ned as the sum of squared residuals of the di erential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-de nite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and veri ed by a set of benchmark problems. The results indicate the ability of the proposed method to e ciently and e ectively solve the linear and non-linear propagation problems.

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