Analyzing Three-Dimensional Laplace Equations Using the Dimension Coupling Method

Fengbin Liu, Mingmei Zuo, Heng Cheng and Ji Ma ()
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Fengbin Liu: College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China
Mingmei Zuo: School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China
Heng Cheng: School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China
Ji Ma: College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China

Mathematics, 2023, vol. 11, issue 17, 1-20

Abstract: Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG) method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has been a longstanding research direction. In this study, we propose the Dimension Coupling Method (DCM) as a promising alternative approach to address this challenge. Based on the Dimensional Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification demonstrates the DCM’s convergence and enhancement of computational speed without losing computational accuracy compared to the IEFG method. Therefore, this proposed method significantly reduces computational time and costs when solving 3D Laplace equations with natural or mixed boundary conditions in a dimensional splitting direction, and expands the applicability of the dimension splitting EFG method.

Keywords: dimension splitting method; improved element-free Galerkin method; dimension coupling method; finite element method; Laplace equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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