The method of fundamental solutions for annular shaped domains

• Published in: Journal of computational and applied mathematics Vol. 228; no. 1; pp. 355 - 372
• Main Author: Li, Zi-Cai
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.06.2009; Elsevier
• Subjects: Acceleration of convergence; Annular shaped domain; Error analysis; Exact sciences and technology; Mathematical analysis; Mathematics; Method of fundamental solutions; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Trefftz method; 65N15; Trefftz method; Annular shaped domain; Error analysis; Method of fundamental solutions; Polynomial; Approximation; Error estimation; Convergence acceleration; Mathematical method; Partial differential equation; Convergence; Fundamental solution; Numerical analysis; Elliptic equation; Boundary value problem; Applied mathematics; Approximation order; Convergence rate; Mathematical model
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2008.09.027

This paper consists of two parts. In the first part, we combine the analysis of Bogomolny [A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM Journal on Numerical Analysis 22 (1985) 644–669], Comodi and Mathon [M.I. Comodi, R. Mathon, A boundary approximation method for fourth order problems, Mathematical Models and Methods in Applied Sciences 1 (1991) 437–445] and Li, et al. [Z.C. Li, R. Mathon, P. Sermer, Boundary methods for solving elliptic equations with singularities, SIAM Journal on Numerical Analysis 24 (1987) 487–498] for the method of fundamental solutions (MFS), and new error bounds are derived for the bounded simply-connected in a readable approach. A factor of O ( n ) is removed in the error bounds of Bogomolny (1985). In the second part, we extend the analysis for the annular domain S a by the MFS. The other fundamental solutions are needed, whose source nodes may also be located uniformly on a circle inside the domain S a , as in the above reference. Error bounds are derived in detail to display the polynomial convergence rates.