Localized Boundary Knot Method for Solving Two-Dimensional Laplace and Bi-Harmonic Equations

Jingang Xiong, Jiancong Wen and Yan-Cheng Liu
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Jingang Xiong: School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
Jiancong Wen: School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
Yan-Cheng Liu: School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China

Mathematics, 2020, vol. 8, issue 8, 1-16

Abstract: In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.

Keywords: localized meshless method; boundary knot method; sparse matrix; Laplace equation; bi-harmonic equation; multiply connected domain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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