Postprocessing Fourier spectral methods: The case of smooth solutions

Applied Numerical Mathematics

Volumn 43, Issue 3, 2002, Pages 191-209

  García Archilla, Bosco ^a   Novo, Julia ^b   Titi, Edriss S ^c  

^a UNIVERSIDAD DE SEVILLA   (Spain)

^b UNIVERSITY OF VALLADOLID   (Spain)

^c University of California at Irvine   (United States)

Author keywords

Aliasing errors; Dissipative equations; Multilevel methods; Spectral methods

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; FOURIER TRANSFORMS; PARTIAL DIFFERENTIAL EQUATIONS;

POSTPROCESSING FOURIER SPECTRAL METHODS;

GALERKIN METHODS;

EID: 0036845619     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(01)00134-9     Document Type: Article

References (48)

1. Bank R.E., Rose D.J. Analysis of multilevel iterative methods for nonlinear finite element equations. Math. Comp. 39:1982;453-465.
2. Brown P.N., Byrne G.D., Hindmarsh A.C. VODE: A variable coefficient ODE Solver. SIAM J. Sci. Statist. Comput. 10:1989;1038-1051.
3. Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. Spectral Methods in Fluid Dynamics. Springer Ser. Comput. Phys. 1988;Springer, Berlin.
4. Cao C., Rammaha M., Titi E.S. The Navier-Stokes equations on the rotating 2 -D sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50:1999;341-360.
5. Collet P., Eckmann J.-P., Epstein H., Stubbe J. Analyticity for the Kuramoto-Sivashinsky equation. Phys. D. 67:1993;321-326.
6. Constantin P., Foias C. Navier-Stokes Equations. Chicago Lectures in Mathematics. 1988;University of Chicago Press, Chicago.
7. Dawson C.N., Wheeler M.F. Two-grid methods for mixed finite element approximations of nonlinear parabolic problems. Contemp. Math. 180:1994;192-204.
8. Debussche A., Marion M. On the construction of families of approximate inertial manifolds. J. Differential Equations. 100:1992;173-201.
9. Devulder C., Marion M. A class of numerical algorithms for large time integration: The nonlinear Galerkin method. SIAM J. Numer. Anal. 29:1992;462-483.
10. Devulder C., Marion M., Titi E.S. On the rate of convergence of nonlinear Galerkin methods. Math. Comp. 60:1993;495-514.
11. Doedel E.J. AUTO: A program for the bifurcation analysis of autonomous systems. Congr. Numer. 30:1981;279-309.
12. Doelman A., Titi E.S. Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation. Numer. Funct. Anal. Optim. 14:1993;299-321.
13. Douglas J. Jr., Dupont T. A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29:1975;696-698.
14. Ferrari A., Titi E.S. Gevrey regularity for nonlinear analytic parabolic equations. Comm. Partial Differential Equations. 23:1998;1-16.
15. Foias C., Jolly M.S., Kevrekidis I.G., Sell G.R., Titi E.S. On the computation of inertial manifolds. Phys. Lett. A. 131:1988;433-436.
16. Foias C., Manley O., Temam R. Modelling of the interaction of small and large eddies in two dimensional turbulent flows. RAIRO Modél. Math. Anal. Numér. 22:1988;93-114.
17. Foias C., Sell G.R., Titi E.S. Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. Dynamics Differential Equations. 1:1989;199-243.
18. Foias C., Temam R. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87:1989;359-369.
19. de Frutos J., García-Archilla B., Novo J. A postprocessed Galerkin method with Chebyshev and Legendre polynomials. Numer. Math. 86:2000;419-442.
20. de Frutos J., Novo J. A spectral element method for the Navier-Stokes equations with improved accuracy. SIAM J. Numer. Anal. 38:2000;799-819.
21. de Frutos J., Novo J. A postprocess-based improvement of the spectral element method. Appl. Numer. Math. 33:2000;217-223.
22. de Frutos J., Novo J. An enhanced pseudospectral Chebyshev method for dissipative partial differential equations. J. Comput. Appl. Math. 115:2000;137-150.
23. García-Archilla B. Some practical experience with the time integration of dissipative equations. J. Comput. Phys. 122:1995;25-29.
24. García-Archilla B., de Frutos J. Time integration of the nonlinear Galerkin method. IMA J. Numer. Anal. 14:1994;1-24.
25. García-Archilla B., Novo J., Titi E.S. Postprocessing the Galerkin method: A novel approach to approximate inertial manifolds. SIAM J. Numer. Anal. 35:1998;941-972.
26. García-Archilla B., Novo J., Titi E.S. An approximate inertial manifold approach to postprocessing Galerkin methods for the Navier-Stokes equations. Math. Comp. 68:1999;893-911.
27. García-Archilla B., Titi E.S. Postprocessing Galerkin methods: The finite-element case. SIAM J. Numer. Anal. 37:2000;470-499.
28. Graham M.D., Steen P.H., Titi E.S. Computational efficiency and approximate inertial manifolds for a Bénard convection system. J. Nonlinear Sci. 3:1993;153-167.
29. Jolly M.S., Kevrekidis I.G., Titi E.S. Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations. Phys. D. 44:1990;38-60.
30. Jolly M.S., Kevrekidis I.G., Titi E.S. Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation. J. Dynamics Differential Equation. 3:1991;179-197.
31. Jones D., Magolin L., Titi E.S. On the effectiveness of the approximate inertial manifolds - Computational study. Theor. Comput. Fluid Dynamics. 7:1995;243-260.
32. Jones D., Titi E.S. A remark on quasi-stationary approximate inertial manifolds for the Navier-Stokes equations. SIAM J. Numer. Anal. 25:1994;894-914.
33. Kreiss H.O. Fourier expansions of the Navier-Stokes equations and their exponential decay rate. Analyse Mathématique et Applications. 1988;245-262 Gauthier-Villars, Paris.
34. Levermore C.D., Oliver M. The complex Ginzburg-Landau equation as a model problem. Lectures in Appl. Math. 31:1996;141-190 American Mathematical Society, Providence, RI.
35. Liu X. Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equations. Phys. D. 50:1991;135-151.
36. Ly H.V., Titi E.S. Global Gevrey regularity for 3-D Bénard convection in porous medium with zero Darcy-Prandtl number. J. Nonlinear Sci. 9:1999;333-362.
37. Mansfield L. On the solution of nonlinear finite element systems. SIAM J. Numer. Anal. 17:1988;752-765.
38. L.G. Margolin, E.S. Titi, S. Wynne, The post-processing Galerkin and nonlinear Galerkin methods - A truncation analysis point of view, submitted.
39. Marion M., Temam R. Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26:1989;1139-1157.
40. Marion M., Xu J. Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32:1995;1170-1184.
41. Novo J., Titi E.S., Wynne S. Efficient methods using high accuracy approximate inertial manifolds. Numer. Math. 87:2001;485-502.
42. Oliver M., Titi E.S. Analyticity of the global attractor and the number of determining nodes for a weakly damped driven nonlinear Schrödinger equation. Indiana Univ. Math. J. 47:1998;49-73.
43. Oliver M., Titi E.S. Gevrey regularity for the attractor of a partially dissipative model of Bénard convection in a porous medium. J. Differential Equation. 163:2000;292-311.
44. Promislow K. Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations. Nonlinear Anal. TMA. 16:1991;959-980.
45. Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci. 68:1988;Springer, Berlin.
46. Titi E.S. On approximate inertial manifolds to the Navier-Stokes equations. J. Math. Anal. Appl. 149:1990;540-557.
47. Xu J. A novel two-grid method for semi-linear elliptic equations. SIAM J. Sci. Comput. 15:1994;231-237.
48. Xu J. Two-grid discretization techniques for linear and nonlinear problems. SIAM J. Numer. Anal. 33:1996;1759-1777.