On the Numerical Solution of Logarithmic Boundary Integral Equations Arising in Laplace's Equations Based on the Meshless Local Discrete Collocation Method
Year: 2019
Author: Pouria Assari, Mehdi Dehghan
Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 4 : pp. 807–837
Abstract
In this article, we investigate the construction of a meshless local discrete collection method suitable for solving a class of boundary integral equations of the second kind with logarithmic singular kernels. These types of boundary integral equations can be deduced from boundary value problems of Laplace's equations with linear Robin boundary conditions. The numerical solution presented in the current paper is obtained by moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The logarithm-like singular integrals appeared in the method are computed via a particular nonuniform Gauss-Legendre quadrature rule. Since the offered scheme is based on the use of scattered points spread on the solution domain and does not require any background meshes, it can be identified as a meshless local discrete collocation (MLDC) method. We also obtain the error bound and the convergence rate of the presented method. The new technique is simple, efficient and flexible for most classes of boundary integral equations. The convergence accuracy of the new technique is examined over four integral equations on various domains and obtained results confirm the theoretical error estimates.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2018-0050
Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 4 : pp. 807–837
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 31
Keywords: Laplace's equation boundary integral equation logarithmic singular kernel discrete collocation method moving least squares (MLS) method error analysis.
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