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;31:16375] 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 < a < 1 ) are obtained by a suitable scaling, where N is the number of admissible functions used. (2) For MFS and CM-RBF, both Cond and Cond_eff are exponential with respect to N, but Cond_eff is much smaller than Cond. (3) For MFS and CM-RBF, the expansion coefficients of FS and radial basis functions may be oscillatingly large, to cause a severe subtraction cancellation in the final solutions. Different behaviours of Cond_eff compared to Cond are explored for different collocation methods in this paper, to provide a comprehensive and intrinsic nature of Cond_eff.
effective condition number condition number collocation trefftz method singularity problem method of fundamental solution