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Stability of discrete orthogonal projections for continuous splines

Published online by Cambridge University Press:  17 April 2009

R.D. Grigorieff  and
I.H. Sloan
R.D. Grigorieff
Affiliation:
Technische Universität Berlin, Strasse des 17. Juni 135, D-10623 Berlin, Germany, e-mail: grigo@math.tu-berlin.de
I.H. Sloan
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia, e-mail: I.Sloan@unsw.edu.au
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Abstract

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In this paper Lp stability and convergence properties of discrete orthogonal projections on the finite element space Sh of continuous polynomial splines of order r are proved. The discrete inner products are defined by composite quadrature rules with positive weights on a sequence of nonuniform grids. It is assumed that the basic quadrature rule Q has at least r quadrature points in order to resolve Sh, but no accuracy is required. The main results are derived under minimal further assumptions, for example the rule Q is allowed to be non-symmetric, and no quasi-uniformity of the mesh is required. The corresponding stability of the orthogonal L2-projections has been studied by de Boor [1] and by Crouzeix and Thomee [2]. Stability of the first derivative of the projection is also proved, under an assumption (unless p = 1) of local quasi-uniformity of the mesh.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 2 , October 1998 , pp. 307 - 332
Copyright
Copyright © Australian Mathematical Society 1998

References

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