Effective condition number for simplified hybrid Trefftz methods

Abstract

The simplified hybrid Trefftz method was first proposed in Trefftz [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7] in 1926 for solving Laplace's equation, where the harmonic functions are chosen as admissible functions, and their linear combination is sought to satisfy the boundary conditions. The error analysis of the hybrid TM is provided in [Li ZC, Chen YL, Georgiou GG, Xenohontos C. Special boundary approximation methods for Laplace equation problems with boundary singularities—applications to the Motz problem. Int Comput Math Appl 2006;51:115–42; Li ZC, Lu TT, Hu HY, Cheng AH-D. Trefftz and collocation methods. Southampton: WIT Publishers; 2007], but no stability analysis exists so far. Also the simplified hybrid techniques have been applied for the TM to couple with the finite element method (FEM) in our previous study and only the error analysis has been made. Hence, the stability analysis is important for the simplified hybrid TM. In this paper, we will apply the effective condition number Cond _ eff . For the original hybrid TM [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7], uniform particular solutions satisfying the governed equation (e.g., the harmonic functions satisfying Laplace's equation) were chosen. In general, piecewise particular solutions can be used for wide application of the hybrid TM, and the interior continuity conditions may be dealt with by hybrid techniques. Their algorithms and error analysis are provided in [Huang HT, Li ZC, Herrera I. Coupling techniques of Trefftz methods. Technical Report, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan; 2006] without stability analysis. In this paper, two cases of the simplified hybrid TM are considered: Case I: uniform particular solutions used, and Case II: piecewise particular solutions used. It is proved that both Cond _ eff and Cond grow exponentially, with respect to the number of particular solutions used. In Case I, Cond is huge and Cond _ eff is significantly smaller than Cond; but in Case II, Cond is moderately large, and Cond _ eff is significantly smaller than Cond. Hence the ill-conditioning of the simplified hybrid TM for Case II is not severe. Such theoretical results have been validated by the numerical experiments. The study of Cond _ eff in this paper provides a complete and comprehensive knowledge of the simplified hybrid TM.