A weak-form RBF-generated finite difference method. (1st May 2020)

• Record Type: Journal Article
• Authors: Jabalameli, Mozhgan; Mirzaei, Davoud
• Is Part Of: Computers & mathematics with applications. Volume 79:issue 9(2020)
• Page Start: 2624
• Page End: 2643
• Publication Date: 2020-05-01
• Subjects: Radial basis functions -- Finite difference method -- Partial differential equations -- Petrov–Galerkin method -- Convergence analysis; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2019.11.024
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, the idea of direct discretization via radial basis functions (RBFs) is applied on a local Petrov–Galerkin test space of a partial differential equation (PDE). This results to a weak-based RBF-generated finite difference (RBF–FD) scheme that possesses some useful properties. The error and stability issues are considered. When the PDE solution or the basis function has low smoothness, the new method gives more accurate results than the already well-established strong-based collocation methods. Although the method uses a Galerkin formulation, it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non-connected, independent and well-shaped subdomains. Some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method.