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Effective Condition Number For Simplified Hybrid Trefftz Methods

机译：简化的混合Trefftz方法的有效条件编号

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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 Condeff 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.
机译：简化的混合Trefftz方法首先在Trefftz中提出[Ein Gegenstuck zum Ritz'schen Verfahren。在：1926年苏黎世第二届应用力学国际会议论文集。 [131-7]在1926年求解拉普拉斯方程，其中选择了谐波函数作为允许函数，并寻求它们的线性组合以满足边界条件。 [Li ZC，Chen YL，Georgiou GG，Xenohontos C提供了混合TM的误差分析。具有边界奇异性的Laplace方程问题的特殊边界逼近方法-应用于Motz问题。 Int Comput Math Appl 2006; 51：115-42;李志诚，卢天天，胡慧云，程AH-D。 Trefftz和搭配方法。南安普敦：WIT出版商； [2007]，但目前尚不存在稳定性分析。此外，在我们先前的研究中，简化的混合技术已应用于TM与有限元方法（FEM）耦合，并且仅进行了误差分析。因此，稳定性分析对于简化的混合动力汽车很重要。在本文中，我们将应用有效条件编号Cond_eff。对于原始的混合动力TM [Ein Gegenstuck zum Ritz'schen Verfahren。在：1926年苏黎世第二届应用力学国际会议论文集。 131-7]，选择满足控制方程式的统一特定解（例如，满足拉普拉斯方程式的谐波函数）。通常，分段特定解决方案可以用于混合TM的广泛应用，并且内部连续性条件可以通过混合技术处理。他们的算法和错误分析在[Huang HT，Li ZC，Herrera I. Trefftz方法的耦合技术中提供。国立中山大学应用数学系技术报告，台湾高雄； [2006]未进行稳定性分析。在本文中，考虑了简化混合TM的两种情况：情况I：使用统一的特定解决方案，情况II：使用分段的特定解决方案。事实证明，相对于所用特定解决方案的数量，Cond_eff和Cond均呈指数增长。在情况I中，Cond很大，而Cond_eff明显小于Cond；但在案例二中，Cond中等偏大，而Condeff明显小于Cond。因此，案例二的简化混合动力TM的病态并不严重。这样的理论结果已经通过数值实验得到了验证。本文对Cond_eff的研究为简化的混合TM提供了完整而全面的知识。

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