Abstract
In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spectral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.
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References
Azaiez M, Shen J, Xu C J, et al. A Laguerre-Legendre spectral method for the Stokes problem in a semi-infinite channel. SIAM J Numer Anal, 2008, 47: 271–292
Babuška I, Guo B Q. Optimal estimates for lower and upper bounds of approximation errors in the p-version of finite element method in two dimensions. Numer Math, 2000, 85: 219–255
Babuška 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 I: Approximability of functions in the weighted Besov spaces. SIAM J Numer Anal, 2001, 39: 1512–1538
Babuška I, Janik T. The h-p version of the finite element method for parabolic equations, Part 1: The p-version in time. Numer Methods Partial Differential Equations, 1989, 5: 363–399
Bernardi C, Coppoletta G, Maday Y. Some spectral approximations of multi-dimensional fourth-order problems. Internal Report 90021. Paris: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 1990
Bernardi C, Dauge M, Maday Y. Sectral Methods for Axisymmetric Domains. In: Ciarlet P G, Lions P L, eds. Series in Appl Math. Paris: Gauhtier-Villars/North-Holland, 1999
Bernardi C, Maday Y. Basic Results on Spectral Methods, R94022. Paris: Université Pierre et Marie Curie, 1994
Bernardi C, Maday Y. Approximations Spectrales de Problèmes aux Limites Elliptiques. Berlin: Springer-Verlag, 1992
Bernardi C, Maday Y. Spectral methods. In: Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis, vol. 5. Techniques of Scientific Computing. Amsterdam: Elsevier, 1997, 209–486
Bernardi C, Maday Y, Rapetti F. Discretisations Variationnelles de Problemes aux Limites Elliptique. Collection: Mathematique et Applications, vol. 45. Berlin: Springer-Verlag, 2004
Boyd J P. Orthogonal rational functions on a semi-infinite interval. J Comp Phys, 1987, 70: 63–88
Boyd J P. Spectral method using rational basis functions on infinite intervals. J Comp Phys, 1987, 69: 112–142
Boyd J P. Chebyshev and Fourier Spectral Methods, 2nd edition. New York: Dover Publication Inc, 2001
Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods in Fluid Dynamics. Berlin: Springer-Verlag, 1988
Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods: Fundamentals in Single Domains. Berlin: Springer-Verlag, 2006
Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Berlin: Springer-Verlag, 2007
Chen Y P, Tang T. Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comp Appl Math, 2009, 233: 938–950
Chen Y P, Tang T. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comp, 2010, 79: 147–167
Christov C I. Complete orthogonal system of functions in L 2(-∞,∞) space. SIAM J Appl Math, 1982, 42: 1337–1344
Coulaud O, Funaro D, Kavian O. Laguerre spectral approximation of elliptic problems in exterior domains. Comp Mech Appl Mech Eng, 1990, 80: 451–458
Dubiner M. Spectral methods on triangles and other domains. J Sci Comput, 1991, 6: 345–390
Everitt W N, Littlejohn L L, Wellman R. The Sobolev orthogonality and spectral analysis of the Laguerre polynomials {L −kn } for positive integers k. J Comp Appl Math, 2004, 171: 199–234
Johnson Fox C M, Guo B Y, Tang T. Combined Hermite spectral-finite difference method for the Fokker-Planck equations. Math Comp, 2001, 71: 1497–1528
Funaro D. Estimates of Laguerre spectral projectors in Sobolev spaces. In: Brezinski C, Gori L, Ronveaux A, eds. Orthogonal Polynomials and Their Applications. New Brunswick: IMACS, 1991
Funaro D. Polynomial Approxiamtions of Differential Equations. Berlin: Springer-Verlag, 1992
Funaro D, Kavian O. Approximation of some diffusion evolution equations in unbounded domains by Hermite functions. Math Comp, 1990, 57: 597–619
Gottlieb D, Orszag S A. Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia: SIAMCBMS, 1977
Gottlieb D, Shu C W. On the Gibbs phenomenon IV: Resolution exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic functions. Math Comp, 1995, 64: 1081–1095
Gottlieb D, Shu C W. Resolution exponential accuracy from collocation point values of piecewise analytic function. Numer Math, 1995, 71: 511–526
Guo B Y. Spectral Methods and Their Applictions. Singapore: World Scientific, 1998
Guo B Y. Gegenbauer approximation and its applications to differential equations on the whole line. J Math Anal Appl, 1998, 22: 180–206
Guo B Y. Error estimation of Hermite spectral method for nonlinear partial differential equations. Math Comp, 1999, 68: 1067–1078
Guo B Y. Jacobi approximation and its applications to differential equations on the half line. J Comp Math, 2000, 18: 95–112
Guo B Y. Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations. SIAM J Numer Anal, 2000, 37: 621–645
Guo B Y. Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J Math Anal Appl, 2000, 243: 373–406
Guo B Y. Gegenbauer approximation and its applications to differential equations with rough asymptotic behaviors at infinity. Appl Numer Math, 2001, 38: 403–425
Guo B Y. Jacobi spectral method for differential equations with rough asymptotic behaviors at infinity. J Comp Math Appl, 2003, 46: 95–104
Guo B Y. Some developments in spectral methods for nonlinear partial differential equations in unbounded domains. In: Gu C H, Hu H S, Li T T, eds. Differential Geometry and Related Topics. Singapore: World Scientific, 2002, 68–90
Guo B Y, Huang W. Mixed Jacobi-spherical harmonic spectral method for Navier-Stokes equations. Appl Numer Math, 2007, 57: 939–961
Guo B Y, Jia H L. Spectral method on quadrilaterals. Math Comp, 2010, 79: 2237–2264
Guo B Y, Jiao Y J. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete Contin Dyn Syst Ser B, 2009, 11: 315–345
Guo B Y, Jiao Y J. Spectral method for Navier-Stokes equations with slip boundary conditions. J Sci Comput, in press, doi: 10.1007/s10915-013-9729-5
Guo B Y, Ma H P. Composite Legendre-Laguerre approximation in unbounded domains. J Comp Math, 2001, 19: 101–112
Guo B Y, Shen J. Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer Math, 2000, 86: 635–654
Guo B Y, Shen J. On spectral approximations using modified Legendre rational functions: application to the Korteweg de Vries equation on the half line. Indiana Univ Math J, 2001, 50: 181–204
Guo B Y, Shen J. Irrational approximations and their applications to partial differential equations in exterior domains. Adv Comp Math, 2008, 28: 237–267
Guo B Y, Shen J, Wang L L. Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J Sci Comput, 2006, 27: 305–322
Guo B Y, Shen J, Wang L L. Generalized Jacobi polynomials/functions and their Applications. Appl Numer Math, 2009, 59: 1011–1028
Guo B Y, Shen J, Wang Z Q. A rational approximation and its applications to differential equations on the half line. J Sci Comput, 2000, 15: 117–148
Guo B Y, Shen J, Wang Z Q. Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Int J Numer Meth Eng, 2002, 53: 65–84
Guo B Y, Shen J, Xu C L. Spectral and pseudospectral approximations using Hermite functions: Application to the Dirac equation. Adv Comp Math, 2003, 19: 35–55
Guo B Y, Shen J, Xu C L. Generalized Laguerre approximation and its applications to exterior problems. J Comp Math, 2005, 23: 113–130
Guo B Y, Sun T, Zhang C. Jacobi and Laguerre quasi-orthogonal approximations and related interpolations. Math Comp, 2013, 82: 413–441
Guo B Y, Wang L L. Jacobi interpolation approximations and their applications to singular differential equations. Adv Comp Math, 2001, 14: 227–276
Guo B Y, Wang L L. Non-isotropic Jacobi spectral method. Contemp Math, 2003 329: 157–164
Guo B Y, Wang L L. Jacobi approximation in non-uniformly Jacobi-weighted Sobolev spaces. J Appro Theor, 2004, 128: 1–41
Guo B Y, Wang L L. Modified Laguerre pseudospectral method refined by multidomain Legendre approximation for differential equations on the half line. J Comp Appl Math, 2006, 190: 304–324
Guo B Y, Wang L L. Error analysis of spectral method on a triangle. Adv Comp Math, 2007, 26: 473–496
Guo B Y, Wang L L, Wang Z Q. Generalized Laguerre interpolation and pseudospectral method for unbounded domains. SIAM J Numer Anal, 2006, 43: 2567–2589
Guo B Y, Wang T J. Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel. Math Comp, 2009, 78: 129–151
Guo B Y, Wang T J. Composite Laguerre-Legendre spectral method for exterior problems. Adv Comp Math, 2010, 32: 393–429
Guo B Y, Wang T J. Composite Laguerre-Legendre spectral method for fourth-order exterior problems. J Sci Comput, 2010, 44: 255–285
Guo B Y, Wang Z Q. Modified Chebyshev rational spectral method for the whole line. Discrete Contin Dyn Syst, 2003, 9: 365–374
Guo B Y, Wang Z Q. Legendre rational approximation on the whole line. Sci China Ser A, 2004, 47: 155–164
Guo B Y, Wang Z Q. Numerical integration based on Laguerre-Gauss interpolation. Comp Meth Appl Mech Eng, 2007, 196: 3726–3741
Guo B Y, Wang Z Q. Legendre-Gauss collocation methods for ordinary differential equations. Adv Comp Math, 2009, 30: 249–280
Guo B Y, Wang Z Q. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete Contin Dyn Syst Ser B, 2010, 14: 1029–1054
Guo B Y, Wang Z Q. A collocation method for generalized nonlinear Klein-Gordon equation. Adv Comp Math, submitted
Guo B Y, Wang Z Q, Tian H J, et al. Integration processes of ordinary differential equations based on Laguerre-Gauss interpolations. Math Comp, 2008, 77: 181–199
Guo B Y, Wang Z Q, Wan Z S, et al. Second order Jacobi approximation with applications to fourth order differential equations. Appl Numer Math, 2005, 55: 480–502
Guo B Y, Yan J P. Legendre-Gauss collocation methods for initial value problems of second order ordinary differential equations. Appl Numer Math, 2009, 59: 1386–1408
Guo B Y, Yi Y G. Generalized Jacobi rational spectral method and its applications. J Sci Comput, 2010, 43: 201–238
Guo B Y, Xu C L. Hermite pseudospectral method for nonlinear partial differential equations. RAIRO Math Model Numer Anal, 2000, 34: 859–872
Guo B Y, Xu C L. Mixed Laguerre-Lagendre pseudospectral method for incompressible fluid flow in an infinite strip. Math Comp, 2003, 73: 95–125
Guo B Y, Yu X H. Composite spectral method for exterior problems with polygonal obstacles. Submitted
Guo B Y, Zhang C, Sun T. Some developments in spectral methods. Stud Adv Math, 2012, 51: 561–574
Guo B Y, Zhang C. Spectral method for high order problems with proper simulations of asymptotic behaviors at infinity. J Comp Appl Math, 2013, 237: 269–294
Guo B Y, Zhang C. Generalized Hermite spectral method matching different asymptotic behaviors at different endpoints. Unpublished
Guo B Y, Zhang K J. On non-isotropic Jacobi pseudospectral method. J Comp Math, 2008, 26: 511–535
Guo B Y, Zhang X Y. A new generalized Laguerre spectral approximation and its applications. J Comp Appl Math, 2005, 181: 342–363
Guo B Y, Zhang X Y. Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions. Appl Numer Math, 2007, 57: 455–471
Hesthaven J S, Gottlieb S, Gottlieb D. Spectral Methods for Time-Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics, vol. 21. Cambridge: Cambridge University Press, 2007
Hesthaven J S, Warburton T. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications. Springer Texts in Applied Mathematics, vol. 54. Berlin: Springer Verlag, 2008
Jia H L, Guo B Y. Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems on polygons. Chinese Ann Math Ser B, 2010, 31: 855–878
Junghanns V P. Uniform convergence of approximate methods for Cauchy type singular equation over (−1, 1). Wiss Z Tech Hocsch Karl-Mars Stadt, 1984, 26: 250–256
Kanyamee N, Zhang Z M. Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems. Int J Numer Anal Model, 2011, 8: 86–104
Karniadakis G E, Sherwin S T. Spectral/hp Element Methods for CFD. Oxford: Oxford University Press, 1999
Li H Y, Shen J. Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle. Math Comp, 2010, 79: 1621–1646
Li H Y, Wang L L. A spectral method on tetrahedra using rational basis functions. Inter J Numer Anal Model, 2010, 7: 330–355
Li X J, Tang T. Convergence analysis of the Jacobi spectral-collocation methods for Abel-Volterra integral equations of the second kind. Front Math China, 2012, 7: 69–84
Li Y Y, Wang L L, Li H Y, et al. A new spectral method on triangles. In: Lecture Notes in Computational Sciences and Engineering. Proceeding of International Conference on Spectral and High-Order Methods (ICOSAHOM09). New York: Springer-Verlag, 2010, 237–246
Lin Y M, Li X J, Xu C J. Finite difference/spectral approximations for the fractional cable equation. Math Comp, 2011, 80: 1369–1396
Ma H P, Guo B Y. Composite Legendre-Laguerre pseudospectral approximation in unbounded domains. IMA J Numer Anal, 2001, 21: 587–602
Ma H P, Sun W W. A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations. SIAM J Numer Anal, 2000, 38: 1425–1438
Ma H P, Sun W W. Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation. SIAM J Numer Anal, 2001, 39: 1380–1394
Ma H P, Sun W W, Tang T. Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains. SIAM J Numer Anal, 2006, 43: 58–75
Ma H P, Zhao T G. A stabilized Hermite spectral method for second-order differential equations in unbounded domain. Numer Methods Partial Differential Equations, 2007, 23: 968–983
Maday Y, Pernaud-Thomas B, Vandeven H. One réhabilitation des méthods spèctrales de type Laguerre. Rech Aérospat, 1985, 6: 353–379
Mastroianni G, Monegato G. Nyström interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations. IMA J Numer Anal, 1997, 17: 621–642
Owens R G. Spectral Approximation on the triangle. Proc R Soc Lond Ser A, 1998, 454: 857–872
Samson M D, Li H Y, Wang L L. A new triangular spectral element method I: implementation and analysis on a triangle. Numer Algor, in press, doi: 10.1007/s11075-012-9677-4
Shen J. Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J Numer Anal, 2000, 38: 1113–1133
Shen J. A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: Application to the KDV equation. SIAM J Numer Anal, 2003, 41: 1595–1619
Shen J, Tang T. Spectral and High-Order Methods with Applications. Beijing: Science Press, 2006
Shen J, Tang T, Wang L L. Spectral Methods: Algorithms, Analysis and Applications. Berlin: Springer Verlag, 2011
Shen J, Wang L L. Error analysis for mapped Legendre spectral and pseudospectral methods. SIAM J Numer Anal, 2004, 42: 326–349
Shen J, Wang L L. Laguerre and composite Legendre-Laguerre dual-Petrov-Galerkin methods for third-order equations. Disc Contin Dyn Syst Ser B, 2006, 6: 1381–1402
Shen J, Wang L L. Legendre and Chebyshev dual-Petrov-Galerkin methods for hyperbolic equations. Comp Meth Appl Mech Eng, 2007, 196: 3785–3797
Shen J, Wang L L. Fourierization of the Legendre-Galerkin method and a new space-time spectral method. Appl Numer Math, 2007, 57: 710–720
Shen J, Wang L L. On spectral approximations in elliptic geometries using Mathieu functions. Math Comp, 2009, 78: 815–844.
Shen J, Wang L L. Some recent advances in spectral methods for unbounded domains. Comm Comp Phys, 2009, 5: 195–241
Shen J, Wang L L. Sparse spectral approximations of the high-dimensional problems based on hyperbolic cross. SIAM J Numer Anal, 2010, 48: 1087–1109
Shen J, Wang L L, Li H Y. A triangular spectral element method using fully tensorial rational basis functions. SIAM J Numer Anal, 2009, 47: 1619–1650
Sherwin S J, Karniadakis G E. A new triangular and tetrahedral basis for high-order finite element methods. Int J Numer Method Eng, 1995, 38: 3775–3802
Stephan E P, Suri M. On the convergence of the p-version of the boundary element Galerkin method. Math Comp, 1989, 52: 31–48
Sun T, Guo B Y. Generalized Jacobi approximation in multiple dimensions and its applications. J Sci Comput, 2013, 55: 327–350
Tang J G, Ma H P. Single and multi-interval Legendre τ-methods in time for parabolic equations. Adv Comput Math, 2002, 17: 349–367
Tang J G, Ma H P. Single and multi-interval Legendre spectral methods in time for parabolic equations. Numer Methods Partial Differential Equations, 2006, 22: 1007–1034
Tang J G, Ma H P. A Legendre spectral method in time for first-order hyperbolic equations. Appl Numer Math, 2007, 57: 1–11
Tang T. The Hermite spectral method for Gaussian type functions. SIAM J Sci Comput, 1993, 14: 594–606
Tang T, Mckee S, Reeks M W. A spectral method for a kinetic equation describing the dispersion of small particles in a turbulent flow. J Comp Phys, 1991, 103: 222–230
Tang T, Xu X, Cheng J. On spectral methods for Volterra type integral equations and the convergence analysis. J Comp Math. 2008, 26: 825–837
Wan Z S, Guo B Y, Wang Z Q. Jacobi pseudospectral method for fourth order problems. J Comp Math, 2006, 24: 481–500
Wang L L, Guo B Y. Jacobi spectral methods for multiple-dimensional singular differential equations. J Comp Math, 2003, 21: 325–338
Wang L L, Guo B Y. Non-isotropic Jacobi spectral methods for unbounded domains. Numer Math, 2004, 13: 204–224
Wang L L, Guo B Y. Stair Laguerre pseudospectral method for differential equations on the half line. Adv Comp Math, 2006, 25: 305–322
Wang L L, Guo B Y. Mixed Fourier-Jacobi spectral method. J Math Anal Appl, 2006, 315: 8–28
Wang L L, Guo B Y. Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature. J Appro Theor, 2009, 161: 142–173
Wang L L, Shen J. Error analysis for mapped Jacobi spectral methods. J Sci Comput, 2005, 24: 183–218
Wang L L, Xie Z Q, Zhao X D. On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math Comp, 2013, 82: 1017–1036
Wang T J, Guo B Y. Mixed Legendre-Hermite pseudospectral method for heat transfer in an infinite plate. J Comp Math, 2005, 23: 587–602
Wang T J, Guo B Y. Composite generalized Laguerre-Legendre pseudospectral method for Fokker-Planck equation in an infinite channel. Appl Numer Math, 2008, 58: 1448–1466
Wang T J, Guo B Y. Composite Laguerre-Legendre pseudospectral method for exterior problems. Comm Comp Phys, 2009, 5: 350–375
Wang T J, Wang Z Q. Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition. Appl Numer Math, 2009, 59: 2444–2451
Wang Z Q, Guo B Y. A rational approximation and its applications to nonlinear differential equations on the whole line. J Math Anal Appl, 2002, 274: 374–403
Wang Z Q, Guo B Y. Modified Legendre rational spectral method for the whole line. J Comp Math, 2004, 22: 457–474
Wang Z Q, Guo B Y. Jacobi rational approximation and spectral method for differential equations of degenerate type. Math Comp, 2008, 77: 883–907
Wang Z Q, Guo B Y. Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations. J Sci Comput, 2012, 52: 226–255
Wang Z Q, Guo B Y, Wu Y N. Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains. Discrete Contin Dyn Syst Ser B, 2009, 11: 1019–1038
Wang Z Q, Guo B Y, Zhang W. Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions. J Comp Appl Math, 2008, 217: 277–298
Wang Z Q, Wang L L. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete Contin Dyn Syst Ser B, 2010, 13: 685–708
Wang Z Q, Wang L L. A collocation method with exact imposition of mixed boundary conditions. J Sci Comput, 2010, 42: 291–317
Xu C L, Guo B Y. Laguerre pseudospectral method for nonlinear partial differential equations. J Comp Math, 2002, 20: 413–428
Xu C L, Guo B Y. Mixed Laguerre-Legendre spectral method for incompressible flow in an infinite strip. Adv Comp Math, 2002, 16: 77–96
Xu C L, Guo B Y. Hermite spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions. Comp Appl Math, 2003, 22: 167–193
Xu C L, Guo B Y. Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions. Appl Math Mech, 2008, 29: 311–331
Xiang X M, Wang Z Q. Generalized Hermite spectral method and its applications to problems in unbounded domains. SIAM J Numer Anal, 2010, 48: 1231–1253
Yan J P, Guo B Y. Laguerre-Gauss collocation method for initial value problems of second-order ODEs. Appl Math Mech, 2011, 32: 1541–1564
Yan J P, Guo B Y. A collocation methods for initial value problems of second order ODEs by using Legendre functions. Numer Math Theo Meth Appl, 2011, 4: 282–294
Yi Y G, Guo B Y. Generalized Jacobi rational spectral method on the half line. Adv Comp Math, 2012, 37: 1–37
Yu X H, Guo B Y. Spectral element method for mixed inhomogeneous boundary value problems of fourth-order. Unpublished
Zhang C, Guo B Y. Domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations on unbounded domains. J Sci Comput, 2012, 53: 451–480
Zhang C, Guo B Y. Generalized Hermite spectral method matching asymptotic behaviors. Submitted
Zhang R, Wang Z Q, Guo B Y. Mixed Fourier-Laguerre spectral and pseudospectral methods for exterior problems using generalized Laguerre functions. J Sci Comput, 2008, 36: 263–283.
Zhang X Y, Guo B Y. Spherical harmonic-generalized Laguerre spectral method for exterior problems. J Sci Comput, 2006, 27: 523–537
Zhang X Y, Guo B Y, Jiao Y J. Spectral method for three-dimensional nonlinear Klein-Gordon equation by generalized Laguerre and spherical harmonic functions. Numer Math Theo Meth Appl, 2009, 2: 43–64
Zhang Z M. Superconvergence points of polynomial spectral interpolation. SIAM J Numer Anal, 2012, 50: 2966–2985
Zhang Z M. Superconvergence of spectral collocation and p-version methods in one dimensional problems. Math Comp, 2005, 74: 1621–1636
Zhang Z M. Superconvergence of a Chebyshev spectral collocation method. J Sci Comput, 2008, 34: 237–246
Zhuang Q Q, Shen J, Xu C J. A coupled Legendre-Laguerre spectral-element method for the Navier-Stokes equations in unbounded domains. J Sci Comput, 2010, 42: 1–22
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Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday
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Guo, B. Some progress in spectral methods. Sci. China Math. 56, 2411–2438 (2013). https://doi.org/10.1007/s11425-013-4660-7
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DOI: https://doi.org/10.1007/s11425-013-4660-7
Keywords
- Jacobi
- Hermite and Laguerre spectral approximations
- Jacobi and Laguerre quasi-orthogonal approximations
- spectral and spectral element methods
- degenerated and singular problems
- problems on nonrectangular and unbounded domains
- problems of non-standard type
- exterior problems