×

zbMATH — the first resource for mathematics

Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. (English) Zbl 1057.41003
The paper is devoted to Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. The approximation results are of the form \[ \left\| Q_N v-v \right\| _{B_1} \leq d_{N,\alpha,\beta}| v |_{B_2}, \] where \(B_1\) and \(B_2\) are non-uniformly Jacobi weighted Sobolev spaces, \(Q_N\) is an orthogonal projection upon the set of polynomials of degree at most \(N\), and \(d_{N,\alpha,\beta}\) is an explicit function of \(N\), \(\alpha\) and \(\beta\), independent of \(v\).
The results are general and all estimates are as sharp as possible. First some basic results on Jacobi approximation are dissussed. Then several orthogonal approximations in non-uniformly Jacobi weighted Sobolev spaces, and also Jacobi-Gauss-type interpolations are studied. These results are useful tools in the numerical solutions of differential and integral equations.

MSC:
41A10 Approximation by polynomials
41A25 Rate of convergence, degree of approximation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Askey, R., Orthogonal polynomials and special functions, Regional conference series in applied mathematics, Vol. 21, (1975), SIAM Philadelphia · Zbl 0298.26010
[2] Babu\(š\)ka, I.; Guo, B.Q., Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two-dimensions, Numer. math., 85, 219-255, (2000) · Zbl 0970.65117
[3] Babu\(š\)ka, I.; Guo, B.Q., Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces, part iapproximability of functions in the weighted Besov spaces, SIAM J. numer. anal., 39, 1512-1538, (2001) · Zbl 1008.65078
[4] Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer Berlin · Zbl 0344.46071
[5] C. Bernardi, M. Dauge, Y. Maday, in: P.G. Ciarlet, P.L. Lions (Eds.), Spectral Methods for Axisymmetric Domains, Series in Applied Mathematics, Vol. 3, Gauhtier-Villars & North-Holland, Paris, 1999. · Zbl 0929.35001
[6] C. Bernardi, Y. Maday, Spectral methods, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, Vol. 5, Techniques of Scientific Computing, Elsevier, Amsterdam, 1997, pp. 209-486.
[7] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer Berlin · Zbl 0658.76001
[8] Canuto, C.; Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. comp., 38, 67-86, (1982) · Zbl 0567.41008
[9] Dubiner, M., Spectral methods on triangles and other domains, J. sci. comput., 6, 345-390, (1991) · Zbl 0742.76059
[10] Ezzirani, A.; Guessab, A., A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations, Math. comp., 225, 217-248, (1999) · Zbl 0916.65020
[11] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, (1977), SIAM-CBMS Philadelphia · Zbl 0412.65058
[12] Gottlieb, D.; Shu, C.W., On the Gibbs phenomenon ivrecovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic functions, Math. comp., 64, 1081-1095, (1995) · Zbl 0852.42018
[13] Guo, Ben-yu, Spectral methods and their applications, (1998), World Scientific Singapore · Zbl 0906.65110
[14] Guo, Ben-yu, Gegenbauer approximation and its applications to differential equations on the whole line, J. math. anal. appl., 226, 180-206, (1998) · Zbl 0913.41020
[15] Guo, Ben-yu, Jacobi spectral approximation and its applications to differential equations on the half line, J. comput. math., 18, 95-112, (2000) · Zbl 0948.65071
[16] Guo, Ben-yu, Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations, SIAM J. numer. anal., 37, 621-645, (2000) · Zbl 0947.65112
[17] Guo, Ben-yu, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. math. anal. appl., 243, 373-408, (2000) · Zbl 0951.41006
[18] Guo, Ben-yu; Shen, Jie; Wang, Zhong-qing, A rational approximation and its applications to differential equations on the half line, J. sci. comp., 15, 117-148, (2000)
[19] Guo, Ben-yu; Shen, Jie; Wang, Zhong-qing, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Internat. J. numer. meth. eng., 53, 65-84, (2002) · Zbl 1001.65129
[20] Guo, Ben-yu; Wang, Li-lian, Jacobi interpolation approximations and their applications to singular differential equations, Adv. comput. math., 14, 227-276, (2001) · Zbl 0984.41004
[21] Ben-yu Guo, Li-lian Wang, Orthogonal approximation on a triangle, unpublished. · Zbl 1116.65122
[22] Junghanns, V.P., Uniform convergence of approximate methods for Cauchy type singular equation over (−1,1), Wiss. Z. tech. hocsch. karl-Mars stadt, 26, 250-256, (1984) · Zbl 0575.65136
[23] Karniadakis, G.; Sherwin, S.J., Spectral/hp element methods for CFD, (1999), Oxford University Press Oxford · Zbl 0954.76001
[24] Kufner, A., Weighted Sobolev spaces, (1985), Wiley New York · Zbl 0567.46009
[25] Schmeisser, H.J.; Triebel, H., Topics in topics in Fourier analysis and function spaces, (1987), Wiley USA · Zbl 0661.46024
[26] Stephan, E.P.; Suri, M., On the convergence of the p-version of the boundary element Galerkin method, Math. comp., 52, 31-48, (1989) · Zbl 0661.65118
[27] Szegö, G., Orthogonal polynomials, (1959), American Mathematical Society Providence, RI · JFM 65.0278.03
[28] Timan, A.F., Theory of approximation of functions of a real variable, (1963), Pergamon Oxford · Zbl 0117.29001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.