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Effective condition number of Trefftz methods for biharmonic equations with crack singularities
Author(s) -
Li ZiCai,
Lu TzonTzer,
Wei Yimin
Publication year - 2009
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.613
Subject(s) - biharmonic equation , mathematics , collocation (remote sensing) , stability (learning theory) , condition number , collocation method , mathematical analysis , boundary value problem , gravitational singularity , polynomial , ordinary differential equation , differential equation , computer science , eigenvalues and eigenvectors , physics , quantum mechanics , machine learning
The paper presents the new stability analysis for the collocation Trefftz method (CTM) for biharmonic equations, based on the new effective condition number Cond_eff. The Trefftz method is a special spectral method with the particular solutions as admissible functions, and it has been widely used in engineering. Three crack models in Li et al . ( Eng. Anal. Boundary Elements 2004; 28 :79–96; Trefftz and Collocation Methods . WIT Publishers: Southampton, Boston, 2008) are considered, and the bounds of Cond_eff and the traditional condition number Cond are derived, to give the polynomial and the exponential growth rates, respectively. The stability analysis explains well the numerical experiments. Hence, the new Cond_eff is more advantageous than Cond. Besides since the bounds of Cond_eff and Cond involve the estimation of the minimal singular value σ min of the discrete matrix F , and since the estimation of σ min is challenging and difficult, the proof for lower bounds of σ min in this paper is important and intriguing. Copyright © 2008 John Wiley & Sons, Ltd.