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26
Some Recent Advances on Spectral Methods for Unbounded Domains
 COMMUNICATIONS IN COMPUTATIONAL PHYSICS
, 2009
"... We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi, Laguerre and Hermite functions. A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out, both theoretic ..."
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Cited by 21 (2 self)
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We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi, Laguerre and Hermite functions. A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out, both theoretically and computationally. A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.
Spectral approximation of the Helmholtz equation with high wave numbers
 SIAM J. Numer. Anal
, 2005
"... Abstract. A complete error analysis is performed for the spectralGalerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green’s functions and does not require any mesh condition ..."
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Cited by 17 (5 self)
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Abstract. A complete error analysis is performed for the spectralGalerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green’s functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multidimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented. Key words. Helmholtz equation, high wave numbers, spectralGalerkin approximation, error analysis
Analysis of a spectralGalerkin approximation to the Helmholtz equation in exterior domains
 SIAM J. Numer. Anal
"... Abstract. An error analysis is presented for the spectralGalerkin method to the Helmholtz equation in 2 and 3dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the DirichlettoNeumann operator, and then a spectralGalerkin method ..."
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Cited by 16 (6 self)
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Abstract. An error analysis is presented for the spectralGalerkin method to the Helmholtz equation in 2 and 3dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the DirichlettoNeumann operator, and then a spectralGalerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented. Key words. Helmholtz equation, wave scattering, error analysis, spectralGalerkin, unbounded domain
CONVERGENCE ANALYSIS OF THE JACOBI SPECTRALCOLLOCATION METHODS FOR VOLTERRA INTEGRAL EQUATIONS WITH A WEAKLY Singular Kernel
, 2009
"... In this paper, a Jacobicollocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard i ..."
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Cited by 16 (4 self)
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In this paper, a Jacobicollocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain highorder accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L∞norm and the weighted L²norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.
A TRIANGULAR SPECTRAL ELEMENT METHOD USING FULLY TENSORIAL RATIONAL BASIS FUNCTIONS
"... Abstract. A rational approximation in a triangle is proposed and analyzed in this paper. The rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform. Optimal error estimates for the L2 − and H1 0 −orthogonal projecti ..."
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Cited by 12 (7 self)
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Abstract. A rational approximation in a triangle is proposed and analyzed in this paper. The rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform. Optimal error estimates for the L2 − and H1 0 −orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle. It is shown that the rational approximation is as accurate as the polynomial approximation in the triangle. Illustrative numerical results, which are in agreement with the theoretical estimates, are presented. Key words. spectral method in a triangle, rational functions, error analysis, spectral element method AMS subject classifications. 65N35, 65N22,65F05, 35J05 1. Introduction. We
ANALYSIS OF SPECTRAL APPROXIMATIONS USING PROLATE SPHEROIDAL WAVE FUNCTIONS
"... Abstract. In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of nonperiodic functions in Sobolev spaces. These results serve as an indispensable tool for ..."
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Cited by 7 (2 self)
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Abstract. In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of nonperiodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectralGalerkin method is proposed and analyzed for elliptictype equations. Illustrative numerical results consistent with the theoretical analysis are also presented. 1.
A Jacobi dualPetrovGalerkin method for solving some oddorder ordinary differential equations,”
 Article ID 947230,
, 2011
"... A Jacobi dualPetrovGalerkin JDPG method is introduced and used for solving fully integrated reformulations of thirdand fifthorder ordinary differential equations ODEs with constant coefficients. The reformulated equation for the Jth order ODE involves nfold indefinite integrals for n 1, . . . ..."
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Cited by 6 (4 self)
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A Jacobi dualPetrovGalerkin JDPG method is introduced and used for solving fully integrated reformulations of thirdand fifthorder ordinary differential equations ODEs with constant coefficients. The reformulated equation for the Jth order ODE involves nfold indefinite integrals for n 1, . . . , J. Extension of the JDPG for ODEs with polynomial coefficients is treated using the JacobiGaussLobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs.
OPTIMAL CONVERGENCE ESTIMATES FOR THE TRACE OF THE POLYNOMIAL L²PROJECTION OPERATOR ON A SIMPLEX
, 2010
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ON EXPONENTIAL CONVERGENCE OF GEGENBAUER INTERPOLATION AND SPECTRAL DIFFERENTIATION
"... Abstract. This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the GegenbauerGauss and GegenbauerGaussLobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in th ..."
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Cited by 2 (2 self)
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Abstract. This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the GegenbauerGauss and GegenbauerGaussLobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximumnormarederived. 1.
ON SPECTRAL APPROXIMATIONS IN ELLIPTICAL GEOMETRIES USING MATHIEU FUNCTIONS
"... Abstract. We consider in this paper approximation properties and applications of Mathieu functions. A first set of optimal error estimates are derived for the approximation of periodic functions by using angular Mathieu functions. These approximation results are applied to study the MathieuLegendre ..."
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Abstract. We consider in this paper approximation properties and applications of Mathieu functions. A first set of optimal error estimates are derived for the approximation of periodic functions by using angular Mathieu functions. These approximation results are applied to study the MathieuLegendre approximation to the modified Helmholtz equation and Helmholtz equation. Illustrative numerical results consistent with the theoretical analysis are also presented. 1.