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Perturbation and stability analysis of strong form collocation with reproducing kernel approximation
Author(s) -
Hu HsinYun,
Chen JiunShyan,
Chi ShengWei
Publication year - 2011
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.3168
Subject(s) - mathematics , condition number , kernel (algebra) , linear system , collocation (remote sensing) , orthogonal collocation , collocation method , exponential function , finite element method , stability (learning theory) , partial differential equation , mathematical analysis , differential equation , ordinary differential equation , computer science , eigenvalues and eigenvectors , physics , quantum mechanics , combinatorics , machine learning , thermodynamics
Solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence, but with the cost of a dense and ill‐conditioned linear system. In this work, the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method, called the reproducing kernel collocation method (RKCM). We perform the perturbation and stability analysis of RKCM, and estimate the condition numbers of the discrete equation. Our stability analyses, validated with numerical tests, show that this approach yields a well‐conditioned and stable linear system similar to that in the finite element method. We also introduce an effective condition number where the properties of both matrix and right‐hand side vector of a linear system are taken into consideration in the measure of conditioning. We first derive the effective condition number of the linear systems resulting from RKCM, and show that using the effective condition number offers a tighter estimation of stability of a linear system. The mathematical analysis also suggests that the effective condition number of RKPM does not grow with model refinement. The numerical results are also presented to validate the mathematical analysis. Copyright © 2011 John Wiley & Sons, Ltd.