Effective Condition Number for Boundary Knot Method

This study makes the first attempt to apply the effective condition number (ECN) to the stability analysis of the boundary knot method (BKM). We find that the ECN is a superior criterion over the traditional condition number. The main difference between ECN and the traditional condition numbers is in that the ECN takes into account the right hand side vector to estimates system stability. Numerical results show that the ECN is roughly inversely proportional to the numerical accuracy. Meanwhile, using the effective condition number as an indicator, one can fine-tune the user-defined parameters (without the knowledge of exact solution) to ensure high numerical accuracy from the BKM.

[1]  Chein-Shan Liu,et al.  A Time-Marching Algorithm for Solving Non-Linear Obstacle Problems with the Aid of an NCP-Function , 2008 .

[2]  W. Chen,et al.  A meshless, integration-free, and boundary-only RBF technique , 2002, ArXiv.

[3]  Jeng-Tzong Chen,et al.  A Modified Method of Fundamental Solutions with Source on the Boundary for Solving Laplace Equations with Circular and Arbitrary Domains , 2007 .

[4]  P. Hansen,et al.  The effective condition number applied to error analysis of certain boundary collocation methods , 1994 .

[5]  Hung-Tsai Huang,et al.  Effective condition number of the Hermite finite element methods for biharmonic equations , 2008 .

[6]  Satya N. Atluri,et al.  A Novel Time Integration Method for Solving A Large System of Non-Linear Algebraic Equations , 2008 .

[7]  Xinrong Jiang,et al.  Investigation of regularized techniques for boundary knot method , 2010 .

[8]  Jichun Li.  Mathematical justification for RBF-MFS , 2001 .

[9]  S. Atluri,et al.  On Solving the Ill-Conditioned System Ax=b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales , 2009 .

[10]  Satya N. Atluri,et al.  A Highly Accurate Technique for Interpolations Using Very High-Order Polynomials, and Its Applications to Some Ill-Posed Linear Problems , 2009 .

[11]  Leevan Ling,et al.  Applicability of the method of fundamental solutions , 2009 .

[12]  D. L. Young,et al.  Novel meshless method for solving the potential problems with arbitrary domain , 2005 .

[13]  Carlos Alberto Brebbia,et al.  Recent advances in boundary element methods , 1978 .

[14]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[15]  Eisuke Kita,et al.  An application of Trefftz method to the sensitivity analysis of two-dimensional potential problem , 1995 .

[16]  Y. C. Hon,et al.  Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems , 2003 .

[17]  T. Chan,et al.  Effectively Well-Conditioned Linear Systems , 1988 .

[18]  D. L. Young,et al.  The Method of Fundamental Solutions for Eigenfrequencies of Plate Vibrations , 2006 .

[19]  Zi-Cai Li,et al.  Effective condition number for simplified hybrid Trefftz methods , 2008 .

[20]  K. H. Chen,et al.  Regularized meshless method for antiplane piezoelectricity problems with multiple inclusions , 2009 .

[21]  Chein-Shan Liu,et al.  Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation , 2008 .

[22]  S. Mukherjee,et al.  THE BOUNDARY NODE METHOD FOR POTENTIAL PROBLEMS , 1997 .

[23]  K. H. Chen,et al.  The Boundary Collocation Method with Meshless Concept for Acoustic Eigenanalysis of Two-Dimensional Cavities Using Radial Basis Function , 2002 .

[24]  Jianming Zhang,et al.  A hybrid boundary node method , 2002 .

[25]  Chein-Shan Liu,et al.  A Fictitious Time Integration Method for Solving Delay Ordinary Differential Equations , 2009 .

[26]  Y. Hon,et al.  The method of fundamental solution for solving multidimensional inverse heat conduction problems , 2005 .