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Superapproximation and commutator properties of discrete orthogonal projections for continuous splines. (English) Zbl 0979.65010

Let \(S_h\) be the space of continuous polynomial splines of order \(r\geq 2\) on the interval \([0, L]\), for a nonuniform mesh. The authors investgate superapproximation and commutator properties of discrete orthogonal projections \(R_h: C[0,L]\to S_h\). Here \(R_h\) is orthogonal with respect to a discrete inner product defined by quadrature rule. The commutator property is a normwise estimate of \((GR_h- R_hG)f\) for arbitrary \(f\in C[0,L]\), where \(G\) is the operator of multiplication with a smooth function.
This paper is based on \(L_p\)-stability and convergence results for \(R_h\) obtained in R. D. Grigorieff and I. H. Sloan [Bull. Aust. Math. Soc. 58, No. 2, 307-332 (1998; Zbl 0926.65015)]. The presented results are similar to those of I. H. Sloan and W. L. Wendland [J. Approx. Theory 97, No. 2, 254-281 (1999; Zbl 0926.41007)] for periodic splines on uniform meshes. As application, the qualocation discretization of the simple equation \(Gu=f\) is considered. Finally, the results are extended to the periodic case.

MSC:

65D07 Numerical computation using splines
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
41A15 Spline approximation
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References:

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