Stability analysis via condition number and effective condition number for the first kind boundary integral equations by advanced quadrature methods, a comparison

Abstract

In our previous study [Huang et al., 2008, 2009, 2010 [21,24,20]; Huang and Lu, 2004 [22,23]; Lu and Huang, 2000 [38]], we have proposed advanced (i.e., mechanical) quadrature methods (AQMs) for solving the boundary integral equations (BIEs) of the first kind. These methods have high accuracy O( h 3), where h = max 1 ⩽ m ⩽ d h m and h m ( m=1,…, d) are the mesh widths of the curved edge Γ m . The algorithms are simple and easy to carry out, because the entries of discrete matrix are explicit without any singular integrals. Although the algorithms and error analysis of AQMs are discussed in Huang et al. (2008, 2009, 2010) [21,24,20], Huang and Lu (2004) [22,23], Lu and Huang (2000) [38], there is a lack of systematic stability analysis. The first aim of this paper is to explore a new and systematic stability analysis of AQMs based on the condition number (Cond) and the effective condition number (Cond_eff) for the discrete matrix K h . The challenging and difficult lower bound of the minimal eigenvalue is derived in detail for the discrete matrix of AQMs for a typical BIE of the first kind. We obtain Cond= O( h min −1) and Cond_eff= O( h min −1), where h min = min 1 ⩽ m ⩽ d h m , to display excellent stability. Note that Cond_eff = O(Cond) is greatly distinct to the case of numerical partial differential equations (PDEs) in Li et al. (2007, 2008, 2009, 2010) [26,31–37], Li and Huang (2008) [27–30], Huang and Li (2006) [19] where Cond_eff is much smaller than Cond. The second aim of this paper is to explore intrinsic characteristics of Cond_eff, and to make a comparison with numerical PDEs. Numerical experiments are carried out for three models with smooth and singularity solutions, to support the analysis made.