On the asymptotic convergence of collocation methods with spline functions of even degree
Authors:
J. Saranen and W. L. Wendland
Journal:
Math. Comp. 45 (1985), 91108
MSC:
Primary 65N99; Secondary 35S99
MathSciNet review:
790646
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Abstract: We investigate the collocation of linear onedimensional strongly elliptic integrodifferential or, more generally, pseudodifferential equations on closed curves by evendegree polynomial splines. The equations are collocated at the respective midpoints subject to uniform nodal grids of the evendegree Bsplines. We prove quasioptimal and optimal order asymptotic error estimates in a scale of Sobolev spaces. The results apply, in particular, to boundary element methods used for numerical computations in engineering applications. The equations considered include Fredholm integral equations of the second and the first kind, singular integral equations involving Cauchy kernels, and integrodifferential equations having convolutional or constant coefficient principal parts, respectively. The error analysis is based on an equivalence between the collocation and certain variational methods with different degree splines as trial and as test functions. We further need to restrict our operators essentially to pseudodifferential operators having convolutional principal part. This allows an explicit Fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing Babuška's stability condition based on strong ellipticity. Our asymptotic error estimates extend partly those obtained by D. N. Arnold and W. L. Wendland from the case of odddegree splines to the case of evendegree splines.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507906463
PII:
S 00255718(1985)07906463
Article copyright:
© Copyright 1985
American Mathematical Society
