## SUPERCONVERGENCE OF SPECTRAL COLLOCATION AND p-VERSION METHODS IN ONE DIMENSIONAL PROBLEMS

Citations: | 5 - 3 self |

### BibTeX

@MISC{Zhang_superconvergenceof,

author = {Zhimin Zhang},

title = {SUPERCONVERGENCE OF SPECTRAL COLLOCATION AND p-VERSION METHODS IN ONE DIMENSIONAL PROBLEMS},

year = {}

}

### OpenURL

### Abstract

Abstract. Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ‖u (k) ‖L ∞ ≤ cM k on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussian points or Lobatto points, the rate of convergence may gain one or two orders of the polynomial degree. 1.

### Citations

395 | Chebyshev and Fourier spectral methods
- Boyd
- 2000
(Show Context)
Citation Context ...ps, the most appreciated property of the spectral method/p-version (finite element) method is the spectral accuracy, geometric/exponential convergent rate. This remarkable behavior is well understood =-=[3, 5, 6, 8, 9, 13, 16, 19, 21]-=-. In the literature, some researchers observed super-geometric convergence rate (see, e.g., [18]) in numerical tests using spectral collocation methods. However, there lacks a theoretical justificatio... |

245 |
Babuska I. Introduction to Finite Element Analysis
- Szabo
- 1989
(Show Context)
Citation Context ...ps, the most appreciated property of the spectral method/p-version (finite element) method is the spectral accuracy, geometric/exponential convergent rate. This remarkable behavior is well understood =-=[3, 5, 6, 8, 9, 13, 16, 19, 21]-=-. In the literature, some researchers observed super-geometric convergence rate (see, e.g., [18]) in numerical tests using spectral collocation methods. However, there lacks a theoretical justificatio... |

223 |
Numerical Analysis of Spectral Methods: Theory and Applications
- Gottlieb, Orszag
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(Show Context)
Citation Context ...ps, the most appreciated property of the spectral method/p-version (finite element) method is the spectral accuracy, geometric/exponential convergent rate. This remarkable behavior is well understood =-=[3, 5, 6, 8, 9, 13, 16, 19, 21]-=-. In the literature, some researchers observed super-geometric convergence rate (see, e.g., [18]) in numerical tests using spectral collocation methods. However, there lacks a theoretical justificatio... |

169 |
Spectral/hp element Methods for CFD
- Karniadakis, Sherwin
- 1999
(Show Context)
Citation Context |

92 | Efficient spectral-Galerkin method I: Direct solvers for the Helmholtz equation and the biharmonic equation using Legendre polynomials
- Shen
- 1994
(Show Context)
Citation Context ...tions in the p-version finite element community since late 1970s; see [19] and references therein. In the early 1990s, Shen introduced them to the spectral method community combined with fast solvers =-=[17]-=-, which made the method more appealing. The counterpart error bound for the hp-version finite element is in the form ( ) p+α eMh ,wherehis the mesh size and α =1, 2. 2p Entire functions and condition ... |

73 |
p- and hp- Finite Element Methods
- Schwab
- 1998
(Show Context)
Citation Context ...aps the most appreciated property of the spectral method/p-version (finite element) method is the spectral accuracy, geometric/exponential convergent rate. This remarkable behavior is well understood =-=[3, 5, 6, 8, 9, 13, 16, 19, 21]-=-. In the literature, some researchers observed supergeometric convergence rate (see, e.g., [18]) in numerical tests using spectral collocation methods. However, a theoretical justification of this phe... |

72 |
Lectures on Entire Functions
- Levin
- 1996
(Show Context)
Citation Context ... |cn| =0. n→∞ n=0 A spectral approximation of f converges at a certain rate depending on the way cn decreases. In order to see the relativeness of condition (M), we need some notation. Here we follow =-=[10]-=-. Define Mf (r) =max |z|=r |f(z)|. If Mf (r) has a “minimum” upper-bound of type eσrρ, then we say that the entire function f is of order ρ and of type σ. Tobemoreprecise, ρ = lim log r Indeed, ρ and ... |

57 | Orthogonal Polynomials, 4th Edition - Szegő - 1975 |

49 |
Spectral methods
- Bernardi, Maday
- 1997
(Show Context)
Citation Context ...aps the most appreciated property of the spectral method/p-version (finite element) method is the spectral accuracy, geometric/exponential convergent rate. This remarkable behavior is well understood =-=[3, 5, 6, 8, 9, 13, 16, 19, 21]-=-. In the literature, some researchers observed supergeometric convergence rate (see, e.g., [18]) in numerical tests using spectral collocation methods. However, a theoretical justification of this phe... |

46 |
Spectral methods for incompressible viscous volume 148 of Applied mathematical sciences
- Peyret
- 2002
(Show Context)
Citation Context |

34 |
Decrypted Secrets: Methods and Maxims of Cryptology
- Bauer
- 1997
(Show Context)
Citation Context ...)] (A.3) n! ≈ ( n e ) n √ 2π(n + 1 6 ). The relative error is less than 1% from n ≥ 3andlessthan0.1% from n ≥ 9. Another useful result is √ (2n)! (A.4) (2n − 1)!! ≈ √ 4 π(n + 1 4 ) , which comes from =-=[2]-=-, (A.5) π = lim n→∞ ( 22n ) ( 2n n ) 2 ( n + 1 ) −1 . 4 Acknowledgments I would like to thank Professor J. Shen at Purdue University for bringing this topic to my attention and for his helpful discuss... |

33 | Orthogonal Functions - Sansone - 1959 |

30 | Interpolation and Approximation by Polynomials - Phillips - 2003 |

15 | A new dual-Petrov-Galerkin method for third and higher odd-order differential equations, application to the KdV equation
- Shen
(Show Context)
Citation Context ...ric/exponential convergent rate. This remarkable behavior is well understood [3, 5, 6, 8, 9, 13, 16, 19, 21]. In the literature, some researchers observed super-geometric convergence rate (see, e.g., =-=[18]-=-) in numerical tests using spectral collocation methods. However, there lacks a theoretical justification of this phenomenon. Observation from interpolation. Let Lk be the Legendre polynomial of degre... |

13 |
The Special Functions and their
- Luke
- 1969
(Show Context)
Citation Context ...)). 5. Final remarks Comparing with Theorem 2.1, there is a completely different strategy for proving the supergeometric rate of convergence for the usual Fourier and Chebyshev expansions; see, e.g., =-=[11]-=-, [8, p. 37], and [4]. The asymptotics of the Legendre polynomials are messier than Fourier or Chebyshev, making the task much harder here. The Legendre polynomials have been used as basis functions i... |

7 |
π unleashed
- Haenel
- 2000
(Show Context)
Citation Context ...(t)dt, k = 1, 2, . . . (1.1) The following properties are valid: OEk+1(x) = 12k + 1 (Lk+1(x) - Lk-1(x)) = 1k(k + 1) (x2 - 1)L0k(x). Zeros of OEk are called Gauss-Lobatto points of degree k. If fp 2 Pp=-=[-1, 1]-=- interpolates a continuous function f at the p + 1 Gauss-Lobatto points: -1 = x0 < x1 < * * * < xp = 1, 1Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. This work was suppor... |

7 |
The rate of convergence of Fourier coefficients for entire functions of infinite order with application to the Weideman-Cloot sinh-mapping for pseudospectral computations on an infinite interval
- Boyd
- 1994
(Show Context)
Citation Context ...Comparing with Theorem 2.1, there is a completely different strategy for proving the super-geometric rate of convergence for the usual Fourier and Chebyshev expansion, see, e.g., [11], [8, p.37], and =-=[4]-=-. The asymptotics of the Legendre polynomials are messier than Fourier or Chebyshev, making the task much harder here. The Legendre polynomials have been used as basis functions in the p-version finit... |

4 |
Applications of Lobatto Polynomials to an Adaptive Finite Element Method: a Posteriori Error Estimates for hp-Adaptivity and Grid-to-Grid
- Moore
(Show Context)
Citation Context ...(x), where ~c can be obtained from T~c = ~wp with T = 0BBB@ 2(,2) 3(,2) * * * p(,2) 2(,3) 3(,3) * * * p(,3). .. ... . . . ... 2(,p) 3(,p) * * * p(,p) 1CCCA . 12sFortunately, T has an explicit inverse =-=[12]-=-: T -1 = 2pp + 1 0BBBB BB@ 1-,22 2p(,2) 2(,2) s2 1-,23 2p(,3) 2(,3) s2 * * * 1-,2p 2p(,p) 2(,p) s2 1-,22 2p(,2) 3(,2) s3 1-,23 2p(,3) 3(,3) s3 * * * 1-,2p 2p(,p) 3(,p) s3. .. ... . . . ... 1-,22 2p(,2... |

3 |
On the hp finite element method for the one dimensional singularly perturbed convection–diffusion problems
- Zhang
(Show Context)
Citation Context ... convergence, superconvergence. This work was supported in part by the National Science Foundation grants DMS-0074301 and DMS-0311807. 1621 c○2005 American Mathematical Society1622 ZHIMIN ZHANG with =-=[22]-=- (1.3) c1(p) =2 p+1 ( ) 2p +2 / ≈ p +1 When f ∈ Cp+1 [−1, 1], the divided difference √ π(p +1) 2 p+1 . f[x0,x1,...,xp,x]= f (p+1)(ξx) , ξx∈ (0, 1). (p +1)! Furthermore, if f satisfies condition (M), w... |

2 |
The Special Functions and their Applications
- Luke
- 1969
(Show Context)
Citation Context ...)). 4. Final Remarks Comparing with Theorem 2.1, there is a completely different strategy for proving the super-geometric rate of convergence for the usual Fourier and Chebyshev expansion, see, e.g., =-=[11]-=-, [8, p.37], and [4]. The asymptotics of the Legendre polynomials are messier than Fourier or Chebyshev, making the task much harder here. The Legendre polynomials have been used as basis functions in... |

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