Error analysis of the method of fundamental solutions for linear elastostatics

• Authors: Zi-Cai Li;Hung-Tsai Huang;Ming-Gong Lee;John Y. Chiang
• Venue: Journal of Computational and Applied Mathematics
• Year: 2013

For linear elastostatics in 2D, the Trefftz methods (i.e., the boundary methods) using the particular solutions and the fundamental solutions satisfying the Cauchy-Navier equation lead to the method of particular solutions (MPS) and the method of fundamental solutions (MFS), respectively. In this paper, the mixed types of the displacement and the traction boundary conditions are dealt with, and both the direct collocation techniques and the Lagrange multiplier are used to couple the boundary conditions. The former is just the MFS and the MPS, and the latter is also called the hybrid Trefftz method (HTM) in Jirousek (1978, 1992, 1996) [1-3]. In Bogomolny (1985) [4] and Li (2009) [5] the error analysis of the MFS is given for Laplace's equation, and in Li (2012) [6] the error bounds of both MPS and HTM using particular solutions (PS) are provided for linear elastostatics. In this paper, our efforts are devoted to explore the error analysis of the MFS and the HTM using fundamental solutions (FS). The key analysis is to derive the errors between FS and PS of the linear elastostatics, where the expansions of the FS in Li et al. (2011) [7] are a basic tool in analysis. Then the optimal convergence rates can be achieved for the MFS and the HTM using FS. Recently, the MFS has been developed with numerous reports in computation; the analysis is behind. The analysis of the MFS for linear elastostatics in this paper may narrow the existing gap between computation and theory of the MFS.