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Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems

机译：奇异问题的惩罚加混合和直接Trefftz方法的稳定性分析

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摘要

For solving the linear algebraic equations Ax = b, the new stability analysis is made based on the effective condition number Cond_eff. The Cond_eff may provide a better upper bound of relative errors of x resulting from the rounding errors of b, than the traditional condition number Cond, which with too large value is, in many times, misleading. In this paper, we apply the effective condition number to the Trefftz methods (TMs) for Poisson's equations with singularities. Two TMs, such as the penalty plus hybrid TM and the Lagrange multiple (i.e., direct) TM, are studied. We focus on the stability analysis of the solutions when the optimal superconvergence is achieved. When solving Motz's problem, the benchmark of singularity problems, by the penalty plus hybrid TM, we have derived that Cond_eff = O(N(2~(1/2))~N) and Cond = O(N~2 2~N), where N is the number of the singular particular functions used. For solving Motz's problem by the direct TM, we have derived that Cond/Cond_eff = O(N (2~(1/2))~N ). Numerical experiments are provided to verify the stability analysis made. In summary, the two TMs are efficient, but the penalty plus hybrid TM is more recommended, due to simplicity of algorithms without extra-variables and less limitations in applications.
机译：为了求解线性代数方程Ax = b，基于有效条件数Cond_eff进行了新的稳定性分析。与传统条件数Cond相比，Cond_eff可以提供由b的舍入误差引起的x相对误差的更好上限，后者在很多情况下都具有太大的误导性。在本文中，我们将有效条件编号应用于具有奇异性的泊松方程的Trefftz方法（TM）。研究了两个TM，例如罚分加混合TM和拉格朗日乘数（即直接）TM。我们专注于实现最佳超收敛时解的稳定性分析。在求解Motz问题，奇异性问题的基准时，通过惩罚加混合TM，我们得出了Cond_eff = O（N（2〜（1/2））〜N）和Cond = O（N〜2 2〜N ），其中N是所使用的单个特殊功能的数量。为了通过直接TM解决Motz问题，我们推导了Cond / Cond_eff = O（N（2〜（1/2））〜N）。提供数值实验以验证所做的稳定性分析。总而言之，这两个TM是有效的，但由于算法简单，没有额外的变量且应用程序的限制较少，因此建议使用惩罚加混合TM。

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