Bibliography

1

M. Avgousti, B. Liu and A. N. Beris, Spectral methods for the viscoelastic time-dependent flow equations with applications to Taylor-Couette flow, International Journal for Numerical Methods in Fluids, 1993, 17, p49-74.

2

B. A. Alpert and V. Rohklin, A fast algorithm for the evaluation of Legendre expansions, SIAM Journal of Scientific Computing, 1991, 12, p158-179.

3

A. Bayliss, T. Belytschko, M. Kulkarni and D. A. Lott-Crumpler, On the dynamics and role of imperfections for localization in thermo-viscoplastic materials, Modelling and Simulation of Materials in Science and Engineering, 1994, 2, p941-964.

4

A. Bayliss, A. Class and B. J. Matkowsky, Adaptive approximation of solutions to problems with multiple layers by Chebyshev pseudo-spectral methods, Journal of Computational Physics, 1995, 116, p160-172.

5

A. Bayliss, A. Class and B. J. Matkowsky, Adaptive pseudo-spectral domain decomposition and the approximation of multiple layers, Journal of Computational Physics, 1995, 119, p132-141.

6

C. Bernardi and G. Coppoletta and Y. Maday, Some spectral approximations of two-dimensional fourth-order problems, Mathematics of Computation, 1992, 59, p63-76.

7

I. Babuška and B.Q. Guo, The h-p version of the finite element method for domains with curved boundaries, SIAM Journal on Numerical Analysis, 1988, 25(4), p837.

8

A. Bayliss, D. Gottlieb, B. J. Matkowsky and M. Minkoff, A. Bayliss and D. Gottlieb and B. Matkowsky and M. Minkoff, An adaptive pseudospectral method for reaction diffusion problems, Journal of Computational Physics, 1989, 81, p421-443.

9

C. Bernardi and A. Karageorghis, Methodes spectrales dans une portione de disque, Numerische Mathematik, 1996.

10

W.Borchers, S.Krautle, R.Pasquetti, R.Rautmann, K.Wielage, C.J.Xu, Towards a parallel hybrid highly accurate Navier-Stokes solver, Numerical Flow Simulation II, NNFM 75, Hirschel Ed., Springer, 2001.

11

C. Bernardi and Y. Maday, Approximations spectrales de problemes aux limites elliptiques Springer-Verlag, Paris, 1992.

12

C. Bernardi and Y. Maday, Spectral Element Methods, ed. P. G. Ciarlet and J. L. Lions, Handbook of Numerical Analysis, North-Holland, Amsterdam, 1996.

13

C. Bernardi and Y. Maday, A collocation method over staggered grids for the Stokes problem, Inter. J. Numer. Meth. Fluids, 1988, 8, p537-557.

14

C. Bernardi and Y. Maday, Properties of some weighted Sobolev spaces and application to spectral approximations, SIAM Journal of Numerical Analysis, 1989, 26, p769-829.

15

I. Babuška and M. Suri, The p and hp versions of the finite element method, basis principles and properties, SIAM Review, 1994, 36, p578-632.

16

W. Couzy and M. O. Deville, A Fast Schur Complement Method for the Spectral Element Discretization of the Incompressible Navier-Stokes Equations, J.Comput. Phys., 1995, 116, p135-142.

17

G. Q. Chen and Q. Du and E. Tadmor, Spectral viscosity approximations to multidimensional scalar conservation laws, Mathematics of Computation, 1993, 61, p629-643.

18

W. Cai and David Gottlieb and C. W. Shu, On one-sided filters for spectral Fourier approximation of discontinuous functions, SIAM Journal of Numerical Analysis, 1992, 29, p905-916.

19

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods for Fluid Dynamics, Springer-Verlag, New York, 1988.

19

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: fundamentals in single domains (Scientific Computation), Springer-Verlag, 2006.

19

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation), Springer-Verlag, 2007.

20

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Mathematics of Computation, 1982, 38, p67-86.

21

C. Canuto and A. Quarteroni, Error estimates for spectral and pseudospectral approximations for hyperbolic equations, SIAM Journal of Numerical Analysis, 1982, 19, p629-642.

22

P. Demaret and M. O. Deville, Chebyshev collocations solutions of the Navier-Stokes equations using multi-domain decomposition and finite element preconditioning, Journal of Computational Physics, 1991, 95, p359-386.

23

W. S. Don and David Gottlieb, The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points, SIAM Journal of Numerical Analysis, 1994, 31, p1519-1534.

24

M. O. Deville and E. H. Mund, Fourier analysis of finite element preconditioned collocation schemes, SIAM Journal of Scientific and Statistical Computing, 1992, 13, p596-610.

25

H. Eisen and W. Heinrichs, A new method of stabilization for singular perturbation problems with spectral methods, SIAM Journal of Numerical Analysis, 1992, 29, p107-122.

26

M. Elghaoui and R. Pasquetti, A spectral embedding method applied to the advection-diffusion equation, Journal of Computational Physics, 1996, 125, p464-476.

27

B. Fornberg, An improved pseudospectral method for initial boundary value problems, Journal of Computational Physics, 1990, 91, p381-397.

28

P. F. Fischer and A. T. Patera, Parallel spectral element solution of the Stokes problem, Journal of Computational Physics, 1991, 92(2), p380-421.

29

A. Frati, F. Pasquarelli and Alfio Quarteroni, Spectral approximation to advection diffusion problems by the fictitious interface method, Journal of Computational Physics, 1993, 107, p201-212.

30

P. F. Fischer, L.-W. Ho, G. E. Karniadakis, E. M. Rønquist and A. T. Patera, Recent advances in parallel spectral element simulation of unsteady incompressible flows, Computers and Structures, 1988, 30, p217-231.

31

D. Funaro, A new scheme for the approximation of advection-diffusion equations by collocation, SIAM Journal of Numerical Analysis, 1993, 30, p1664-1676.

31

V. Girault, P.-A. Raviart, Finite element methods for the Navier-Stokes equations: Theory and algorithms, Springer-Verlag, Berlin, 1986.

32

D. Funaro, Spectral Elements for Transport-Dominated Equations, Lecture Notes in Computational Science and Engineering, Springer-Verlagi, Berlin, 1997, 1.

33

D. Gottlieb and J.S. Hesthaven, Spectral methods for hyperbolic problems , Journal of Computational and Applied Mathematics, 2001, 128, p83-131.

34

D. Gottlieb and R. S. Hirsh, , Journal of Scientific Computing, 1989, 4(1), p309-326.

35

David Gottlieb and M. Y. Hussaini and Steven A. Orszag, Theory and application of spectral methods, ed. G. Voigt and D. Gottlieb and M. Y. Hussaini, Spectral Methods for Partial Differential Equations, Part,p1-54, SIAM, Philadelphia, 1984.

36

B.-Y. Guo and H.-P. Ma, A pseudospectral-finite element method for solving two-dimensional vorticity equations, SIAM Journal of Numerical Analysis, 1991, 28, p113-132.

37

B.-Y. Guo and H.-P. Ma, Combined finite element and pseudospectral methods for the two-dimensional evolutionary Navier-Stokes equations, SIAM Journal of Numerical Analysis, 1993, 30, p1066-1083.

38

D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon V: Recovering exponential accuracy from collocation point values of a piecewise analytic function, Numerische Mathematik, 1995, 71, p511-526.

39

D. Rh. Gwynllyw and A. R. Davies and T. N. Phillips, A Moving Spectral Element Approach to the Dynamically Loaded Bearing Problem, J.Comput. Phys., 1996, 123, p476-494.

40

W. Heinrichs, Splitting techniques for the pseudospectral approximation of the unsteady Stokes equation, SIAM Journal of Numerical Analysis, 1993, 30, p19-39.

41

W. Heinrichs, Spectral multi-grid techniques for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 1993, 106, p297-314.

42

W. Heinrichs, Spectral multigrid methods for the reformulated Stokes equations, Journal of Computational Physics, 1993, 107(2), p213-224.

43

W. Heinrichs, Spectral methods for singular perturbation problems, Applications of Mathematics, 1994, 39(3), p161-188.

44

W. Heinrichs, Spectral multigrid methods for domain decomposition problems using patching techniques, Applied Mathematics and Computation, 1993, 59(2), p165-176.

45

W. Heinrichs, Defection correction for the advection-diffusion equation, Computer Methods in Applied Mechanics and Engineering, 1994, 119, p191-197.

46

R. D. Henderson, Unstructured Spectral Element Methods: Parallel Algorithms and Simulations, Princeton University, 1994.

47

R. D. Henderson, Spectral Elements: Adaptivity and Applications in Fluid Dynamics, Int. Conf. Comp. Eng. Sci., 1995, Hawaii.

48

R. Hunt, Three-dimensional steady flow in a dividing channel using finite and pseudospectral differences, SIAM Journal of Scientific Computing, 1996, 17(3), p561-578.

49

J. S. Hesthaven and D. Gottlieb, A stable penalty method for the compressible Navier-Stokes equations. I. Open boundary conditions, SIAM Journal of Scientific Computing, 1996, 17(3), p579-612.

50

Ron D. Henderson and George Em Karniadakis, Hybrid Spectral-Element-Low-Order Methods for Incompressible Flows, J.Sci. Comp., 1991, 6(2), p79-100.

51

R. D. Henderson and G. E. Karniadakis, Unstructured Spectral Element Methods for the Incompressible Navier-Stokes Equations, 8th Intnl Conf. Finite Element Methods in Fluids, 1993, Barcelona.

52

R. D. Henderson and G. E. Karniadakis, Unstructured Spectral Element methods for Simulation of Turbulent Flows, J.Comput. Phys., 1995, 122, p191-217.

53

L.-W. Ho and E. M. Rønqusit, Spectral Element Solution of Steady Incompressible Viscous Free-Surface Flows, Finite Elements Anal. Des., 1994, 16, p207-227.

54

W.-Z. Huang and D. M. Sloan, A new pseudospectral method with upwind features, IMA Journal of Numerical Analysis, 1993, 13, p413-430.

55

M. Iskandarani and D. B. Haidvogel and J. P. Boyd, A Staggered Spectral Element Model with Application to the Oceanic Shallow Water Equations, Int. J. for Num. Meth. Fluids, 1995, 20, p393-414.

56

G. Ierley, B. Spencer and R. Worthing, Spectral methods in time for a class of parabolic partial differential equations, Journal of Computational Physics, 1992, 102, p88-97.

57

G. E. Karniadakis, Spectral Element Simulations of Laminar and Turbulent Flows in Complex Geometries, Appl. Num. Math., 1989, 6, p85-105.

58

G. E. Karniadakis, Spectral Element-Fourier Methods for Incompressible Turbulent Flows, Comp. Meth. Appl. Mech. & Engng, 1990, 80, p367-380.

59

G. E. Karniadakis and M. Israeli and S. A. Orszag, High-Order Splitting Methods for the Incompressible Navier-Stokes Equations, J.Comput. Phys., 1991, 97(2), p414-443.

60

G. E. Karniadakis and S. A. Orszag, Nodes, Modes and Flow Codes, Physics Today, 1993, p34-42.

61

D. A. Kopriva and J. H. Kolias, A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows, J.Comput. Phys., 1996, 125, p244-261.

62

D. A. Kopriva, A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows. II. A Semi-Structured Method, J.Comput. Phys., 1996, 128, p475-488.

63

K. Z. Korczak and A. T. Patera, An Isoparametric Spectral Element Method for Solution of the Navier-Stokes Equations in Complex Geometry, J.Comput. Phys., 1986, 62, p361-382.

64

Y. Maday, S. M. Ould Kaber and E. Tadmor, Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM Journal of Numerical Analysis, 1989, 312, p705-710.

65

Y. Maday and D. Meiron and A. T. Patera and E. M. Rønquist, Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations, SIAM Journal of Scientific Computing, 1993, 14, p310-337.

66

Y. Maday and A. T. Patera and E. M. Rønquist, An operator integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow, Journal of Scientific Computing, 1990, 5, p263-292.

67

Y. Maday and E. M. Rønquist, Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries, Computer Methods in Applied Mechanics and Engineering, 1990, 80, p91-115.

68

C. Mavriplis, Adaptive Mesh Strategies for the Spectral Element Method, Comp. Meth. Appl. Mech. & Engng, 1994, 116, p77-86.

69

Y. Maday and A. T. Patera, Spectral Element Methods for the Incompressible Navier-Stokes Equations, 3, pp71-143, ASME, 1989. ed. A. K. Noor and J. T. Oden., State-of-the-Art Surveys on Computational Mechanics.

70

J. M. Occhialini and G. P. Muldowney and J. J. L. Higdon, Boundary Integral/Spectral Element Approaches to the Navier-Stokes Equations, Int. J. for Num. Meth. Fluids, 1992, 15, p1361-1381.

71

A. T. Patera, A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion, J.Comput. Phys., 1984, 54, p468-488.

72

A. Pinelli, W. Couzy, Michel O. Deville and C. Benocci, An efficient iterative solution method for the Chebyshev collocation of advection-dominated transport problems, SIAM Journal of Scientific Computing, 1996, 17(3), p647-657.

73

D. Pathria and G. E. Karniadakis, Spectral Element Method for Elliptic Problems in Nonsmooth Domains, J.Comput. Phys., 1995, 122, p83-95.

74

T. N. Phillips and A. Karageorghis, Efficient direct methods for solving the spectral collocation equations for Stokes flow in rectangularly decomposable domains, SIAM Journal of Scientific and Statistical Computing, 1989, 10, p89-103.

75

R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow, Springer-Verlag, New York, 1983.

76

L. F. Pavarino and O. B. Widlund, A polylogarithmic bound for an iterative substructuring method for spectral elements, SIAM Journal of Numerical Analysis, 1996, 33, p1303-1335.

77

R.Pasquetti, C.J.Xu, Note: Comments on Filter-Based Stabilization of Spectral Element Methods, J. Comput. Phys., 2002, 182, p646-650.

78

R.Pasquetti, C.J.Xu, High-Order Algorithms for Large-Eddy Simulation of incompressible Flows, J. Scient. Computing, 2002, 17(1-3).

79

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, 23, Springer Series in Computational Mathematics, Springer-Verlag, Heidelberg, 1994.

80

A. Quarteroni and E. Zampieri, Finite-element preconditioning for Legendre spectral collocation approximations to elliptic-equations and systems, SIAM Journal of Numerical Analysis, 1992, 29, p917-936.

81

E. M. Ronquist, Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier-Stokes Equations, Massachusetts Institute of Technology, 1988.

82

E. M. Rønquist, Convection Treatment Using Spectral Elements of Different Order, Int. J. for Num. Meth. Fluids, 1996, 22, p241-264.

83

P. E. Raad and A. Karageorghis, A Chebyshev spectral collocation method for the solution of the Reynolds equation of lubrication, Journal of Computational Physics, 1993, 106, p42-51.

84

R. D. Russell, D. M. Sloan and M. R. Trummer, On the structure of Jacobians for spectral methods for nonlinear partial differential equations, SIAM Journal of Scientific and Statistical Computing, 1992, 13(2), p541-549.

85

J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM Journal of Scientific Computing, 1994, 15(6), p1489-1505.

86

J. Shen, On fast Poisson solver, inf-sup constant and iterative Stokes solver by Legendre Galerkin method, Journal of Computational Physics, 1995, 116, p184-188.

87

C. W. Shu, G. Erlebacher, T. A. Zang, D. Whitaker and S. Osher, High-order ENO schemes applied to two- and three-dimensional compressible flow, Applied Numerical Mathematics, 1992, 9, p45-71.

88

S. J. Sherwin and G. E. Karniadakis, Tetrahedral hp Finite Elements: Algorithms and Flow Solutions, J.Comput. Phys., 1996, 124, p14-45.

89

J. Shen and R. Témam, Nonlinear Galerkin methods using Chebyshev or Legendre polynomials I. One dimensional case, SIAM Journal of Numerical Analysis, 1995, 32, p215-234.

90

D. M. Sloan and R. Wallace, Pseudospectral solution of a nonlinear dissipative system, SIAM Journal of Scientific Computing, 1995, 16, p1049-1070.

91

E. Süli and A. Ware, A spectral method of characteristics for hyperbolic problems, SIAM Journal of Numerical Analysis, 1991, 28, p423-445.

92

E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM Journal of Numerical Analysis, 1989, 26, p30-44.

93

E. Tadmor, Shock capturing by the spectral viscosity method, Computer Methods in Applied Mechanics and Engineering, 1990, 80, p197-208.

94

H. Tal-Ezer, A pseudospectral Legendre method for hyperbolic equations with an improved stability condition, Journal of Computational Physics, 1986, 67, p145-167.

95

H. Tal-Ezer, Spectral methods in time for parabolic problems, SIAM Journal of Numerical Analysis, 1989, 26, p1-11.

96

T. Tang, The Hermite spectral method for Gaussian-type functions, SIAM Journal of Scientific Computing, 1993, 14, p594-606.

97

T. Tang and M. R. Trummer, Boundary layer resolving pseudospectral methods for singular perturbation problems, SIAM Journal of Scientific Computing, 1995, 17, p430-438.

98

L. N. Trefethen and M. R. Trummer, An instability phenomena in spectral methods, SIAM Journal of Numerical Analysis, 1987, 24, p1008-1023.

99

H. Vandeven, Family of spectral filters for discontinuous problems, SIAM Journal of Scientific Computing, 1991, 6, p159-192.

100

L. Vozovoi, M. Israeli and A. Averbuch, Spectral multi-domain technique with Local Fourier Basis II: decomposition into cells SIAM Journal of Scientific Computing, 1994, 9, p311-326.

101

J. Andre C. Weideman, The eigenvalues of Hermite and rational spectral differentiation matrices, Numerische Mathematik, 1992, 61, p409-431.

102

J. Andre C. Weideman and A. Cloot, Spectral methods and mappings for evolution equations on the infinite line, Computer Methods in Applied Mechanics and Engineering, 1990, 80, p467-481.

103

D. C. Witte and P. G. Richards, The pseudospectral method for simulating wave propagation, Computational Acoustics, 1990, 3, p1-18.

104

C.J. Xu, An efficient method for the Navier-Stokes/Euler coupled equations via a spectral element approximation, SIAM J. Numer. Anal., 2000, 38(4), 1217-1242.

105

C.J. Xu, Y.M. Lin, Open boundary conditions for the spectral simulation of Poiseuille-Benard flow, Acta mechanica sinica, 2000, 32(1), p1-10.

106

C.J.Xu, Y.M. Lin, Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation, Inter. J Numer. Meth. Fluids, 2000, 32,619-646.

107

C.J.Xu, Y.Maday, A global algorithm in spectral method for viscous/in-viscid coupling, Chinese Annual of Math.(B), 1997, 18(2), p191-200.

108

C.J.Xu, R.Pasquetti, On the efficiency of the semi-Lagrangian method for the incompressible flow, Inter. J Numer. Meth. Fluids (International Journal for Numerical Methods in Fluids), 2001, 35,319-340.

109

T. A. Zang, Spectral methods for simulations of transition and turbulence, Computers Methods in Applied Mechanics and Engineering, 1990, 80, p209-221.

110

Y. Zhang, J. I. D. Alexander and J. Ouazzani, A Chebyshev collocation method for moving boundaries, heat transfer, and convection during directional solidification, International Journal for Numerical Methods for Heat and Fluid Flow, 1994, 4, p115-129.