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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: [email protected]
I.H. Sloan
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia, e-mail: [email protected]
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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

[1]de Boor, C., ‘A bound on the L ∞-norm of L 2-approximation by splines in term of a global mesh ratio’, Math. Comp. 30 (1976), 765–771.Google Scholar
[2]Crouzeix, M. and Thomée, V., ‘The stability in L p and of the L 2-projection onto finite element function spaces’, Math. Comp. 48 (1987), 521–532.Google Scholar
[3]Davis, P.J. and Rabinowitz, P., Methods of numerical integration (Academic Press, New York, 1975).Google Scholar
[4]Descloux, J., ‘On finite element matrices’, SIAM J. Numer. Anal. 9 (1972), 260–265.CrossRefGoogle Scholar
[5]Douglas, J., Dupont, T. and Wahlbin, L., ‘Optimal L ∞ error estimates for Galerkin approximations to solutions of two-point boundary value problems’, Math. Comp. 29 (1975), 475–483.Google Scholar
[6]Douglas, J., Dupont, T. and Wahlbin, L., ‘The stability in L q of the L 2-projection into finite element subspaces’, Numer. Math. 23 (1975), 193–197.CrossRefGoogle Scholar
[7]Dautray, R. and Lions, J.-L., Mathematical analysis and numerical methods for science and technology 4 (Springer-Verlag, Berlin, Heidelberg, New York, 1990).Google Scholar
[8]Ganesh, M. and Sloan, I.H., ‘Optimal order spline methods for nonlinear differential and integro-differential equations’, Appl. Numer. Math, (to appear).Google Scholar
[9]Grigorieff, R.D. and Sloan, I.H., ‘Spline Petrov-Galerkin methods with quadrature’, Numer. Fund. Anal. Optim. 17 (1996), 755–784.CrossRefGoogle Scholar
[10]Grigorieff, R.D. and Sloan, I.H., ‘High-order spline Petrov-Galerkin methods with quadrature’, in Proceedings of the International Congress on Industrial and Applied Mathematics Hamburg, 1995,, (Alefeld, G. et al. , Editors) (Z. Angew Math. Mech. 76, 1996), pp. 15–18.Google Scholar
[11]Wahlbin, L., Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics 1605 (Springer, Berlin, 1995).CrossRefGoogle Scholar