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Superconvergence of a Chebyshev spectral collocation method
 J. Sci. Comput
, 2008
"... Abstract We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) forthe twopoint boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind ..."
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Abstract We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) forthe twopoint boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Supergeometric convergent rate is established for a special class of solutions.
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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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.
ANY ORDER SUPERCONVERGENCE FINITE VOLUME SCHEMES FOR 1D GENERAL ELLIPTIC EQUATIONS
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A POSTERIORI ERROR ESTIMATION FOR hpADAPTIVITY FOR FOURTHORDER EQUATIONS
"... Abstract. A posteriori error estimates developed to drive hpadaptivity for secondorder reactiondiffusion equations are extended to fourthorder equations. A C1 hierarchical finite element basis is constructed from HermiteLobatto polynomials. A priori estimates of the error in several norms for b ..."
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Abstract. A posteriori error estimates developed to drive hpadaptivity for secondorder reactiondiffusion equations are extended to fourthorder equations. A C1 hierarchical finite element basis is constructed from HermiteLobatto polynomials. A priori estimates of the error in several norms for both the interpolant and finite element solution are derived. In the latter case this requires a generalization of the wellknown AubinNitsche technique to timedependent fourthorder equations. We show that the finite element solution and corresponding HermiteLobatto interpolant are asymptotically equivalent. A posteriori error estimators based on this equivalence for solutions at two orders are presented. Both are shown to be asymptotically exact on grids of uniform order. These estimators can be used to control various adaptive strategies. Computational results for linear steadystate and timedependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings. 1.
Computing and Information COMPARISON OF A SPECTRAL COLLOCATION METHOD AND SYMPLECTIC METHODS FOR HAMILTONIAN SYSTEMS
"... Abstract. We conduct a systematic comparison of a spectral collocation method with some symplectic methods in solving Hamiltonian dynamical systems. Our main emphasis is on nonlinear problems. Numerical evidence has demonstrated that the proposed spectral collocation method preserves both energy an ..."
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Abstract. We conduct a systematic comparison of a spectral collocation method with some symplectic methods in solving Hamiltonian dynamical systems. Our main emphasis is on nonlinear problems. Numerical evidence has demonstrated that the proposed spectral collocation method preserves both energy and symplectic structure up to the machine error in each time (large) step, and therefore has a better long time behavior.
doi:10.4208/jcm.2009.10m1006 SUPERGEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS *
"... We propose and analyze a C 0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A supergeometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptoti ..."
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We propose and analyze a C 0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A supergeometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a GaussLobatto collocation method and a spectral Galerkin method is established for a simplified model.
SHARP ERROR BOUNDS FOR JACOBI EXPANSIONS AND GENGENBAUERGAUSS QUADRATURE OF ANALYTIC FUNCTIONS
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