Study on effective condition number for collocation methods

In this paper, the stability study is further made for three kinds of collocation methods. (I) The general collocation Trefftz methods (GCTMs) using piecewise particular solutions, while in Li et al. [Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems. Eng Anal Boundary Elem 2004;31:163–75] the collocation Trefftz methods (CTMs) using the uniform particular solutions in the entire domain. (II) The method of fundamental solutions (MFSs), i.e., the CTM using the fundamental solution (FS). (III) The collocation method using the radial basis functions (CM-RBFs). In this paper, the new bounds of the traditional condition number Cond and the effective condition number Cond_eff are derived, and comparisons are made to confirm that the effective condition number is a better criterion for stability analysis. The main results are as follows. (1) For Motz's problem by GCTM, Cond_eff=O(N)Cond_eff=O(N) and Cond=O(a-N)(0