caa a22 4500 47577700X CHVBK 20180406123633.0 cr unu---uuuuu 170329e20000901xx s 000 0 eng 10.1023/A:1011126400782 doi (NATIONALLICENCE)springer-10.1023/A:1011126400782 Gelb Anne Department of Mathematics, Arizona State University, P.O. Box 871804, 85287-1804, Tempe, Arizona aut A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions [Elektronische Daten] [Anne Gelb] Consider a piecewise smooth function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Various types of reconstruction methods developed for the recovery of piecewise smooth functions have met with varying degrees of success. The Gegenbauer reconstruction method, originally proposed by Gottlieb et al. has the particularly impressive ability to reconstruct piecewise analytic functions with exponential convergence up to the points of discontinuity. However, it has been sharply criticized for its high cost and susceptibility to round-off error. In this paper, a new approach to Gegenbauer reconstruction is considered, resulting in a reconstruction method that is less computationally intensive and costly, yet still enjoys superior convergence. The idea is to create a procedure that combines the well known exponential filtering method in smooth regions away from the discontinuities with the Gegenbauer reconstruction method in regions close to the discontinuities. This hybrid approach benefits from both the simplicity of exponential filtering and the high resolution properties of the Gegenbauer reconstruction method. Additionally, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansions is introduced that is both more cost effective and less prone to round-off errors. Plenum Publishing Corporation, 2000 Fourier expansion nationallicence Gibbs phenomenon nationallicence piecewise smoothness nationallicence reconstruction nationallicence Journal of Scientific Computing Kluwer Academic Publishers-Plenum Publishers 15/3(2000-09-01), 293-322 0885-7474 15:3<293 2000 15 10915 https://doi.org/10.1023/A:1011126400782 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1011126400782 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Gelb Anne Department of Mathematics, Arizona State University, P.O. Box 871804, 85287-1804, Tempe, Arizona aut NATIONALLICENCE 773 0- Journal of Scientific Computing Kluwer Academic Publishers-Plenum Publishers 15/3(2000-09-01), 293-322 0885-7474 15:3<293 2000 15 10915 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer