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Quadratic‐spline collocation methods for two‐point boundary value problems
Author(s) -
Houstis E. N.,
Christara C. C.,
Rice J. R.
Publication year - 1988
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.1620260412
Subject(s) - mathematics , superconvergence , midpoint , quadratic equation , boundary value problem , second derivative , collocation (remote sensing) , mathematical analysis , collocation method , spline (mechanical) , geometry , finite element method , differential equation , ordinary differential equation , physics , remote sensing , thermodynamics , geology
A new collocation method based on quadratic splines is presented for second order two‐point boundary value problems. First, O ( h 4 ) approximations to the first and second derivative of a function are derived using a quadratic‐spline interpolant of u. Then these approximations are used to define an O ( h 4 ) perturbation of the given boundary value problem. Second, the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal O ( h 3‐J ) global estimate for the j th derivative of the error is derived. Further, O ( h 4‐J ) error bounds for the j th derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O ( h 2‐J ) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method.