An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations
HTML articles powered by AMS MathViewer
- by Bosco García-Archilla, Julia Novo and Edriss S. Titi PDF
- Math. Comp. 68 (1999), 893-911 Request permission
Abstract:
In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.References
- Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
- Ali Ait Ou Ammi and Martine Marion, Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations, Numer. Math. 68 (1994), no. 2, 189–213. MR 1283337, DOI 10.1007/s002110050056
- Randolph E. Bank and Donald J. Rose, Analysis of a multilevel iterative method for nonlinear finite element equations, Math. Comp. 39 (1982), no. 160, 453–465. MR 669639, DOI 10.1090/S0025-5718-1982-0669639-X
- Paul F. Batcho and George Em. Karniadakis, Generalized Stokes eigenfunctions: a new trial basis for the solution of incompressible Navier-Stokes equations, J. Comput. Phys. 115 (1994), no. 1, 121–146. MR 1300335, DOI 10.1006/jcph.1994.1182
- H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), no. 4, 677–681. MR 582536, DOI 10.1016/0362-546X(80)90068-1
- Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259, DOI 10.7208/chicago/9780226764320.001.0001
- Clint N. Dawson and Mary F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 191–203. MR 1312392, DOI 10.1090/conm/180/01971
- A. Debussche and M. Marion, On the construction of families of approximate inertial manifolds, J. Differential Equations 100 (1992), no. 1, 173–201. MR 1187868, DOI 10.1016/0022-0396(92)90131-6
- Christophe Devulder and Martine Marion, A class of numerical algorithms for large time integration: the nonlinear Galerkin methods, SIAM J. Numer. Anal. 29 (1992), no. 2, 462–483. MR 1154276, DOI 10.1137/0729028
- Christophe Devulder, Martine Marion, and Edriss S. Titi, On the rate of convergence of the nonlinear Galerkin methods, Math. Comp. 60 (1993), no. 202, 495–514. MR 1160273, DOI 10.1090/S0025-5718-1993-1160273-1
- Jim Douglas Jr. and Todd Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp. 29 (1975), 689–696. MR 431747, DOI 10.1090/S0025-5718-1975-0431747-2
- T. Dubois, F. Jauberteau, M. Marion, and R. Temam, Subgrid modelling and the interaction of small and large wavelengths in turbulent flows, Comput. Phys. Comm. 65 (1991), no. 1-3, 100–106. MR 1104966, DOI 10.1016/0010-4655(91)90160-M
- C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, and E. S. Titi, On the computation of inertial manifolds, Phys. Lett. A 131 (1988), no. 7-8, 433–436. MR 972615, DOI 10.1016/0375-9601(88)90295-2
- C. Foias, O. Manley, and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 1, 93–118 (English, with French summary). MR 934703, DOI 10.1051/m2an/1988220100931
- Ciprian Foias, George R. Sell, and Roger Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), no. 2, 309–353. MR 943945, DOI 10.1016/0022-0396(88)90110-6
- Ciprian Foias, George R. Sell, and Edriss S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynam. Differential Equations 1 (1989), no. 2, 199–244. MR 1010966, DOI 10.1007/BF01047831
- C. Foias and R. Temam, The algebraic approximation of attractors: the finite-dimensional case, Phys. D 32 (1988), no. 2, 163–182. MR 969028, DOI 10.1016/0167-2789(88)90049-8
- J. de Frutos, B. García-Archilla and J. Novo, A postprocessed Galerkin method with Chebyshev and Legendre polynomials, (submitted).
- B. García-Archilla, Some practical experience with the time integration of dissipative equations J. Comp. Phys., 122, 1995, 25–29.
- Bosco García-Archilla and Javier de Frutos, Time integration of the non-linear Galerkin method, IMA J. Numer. Anal. 15 (1995), no. 2, 221–244. MR 1323739, DOI 10.1093/imanum/15.2.221
- B. García-Archilla, J. Novo and E. S. Titi, Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds, SIAM J. Numer. Anal. 35, 1998, 941–972.
- B. García-Archilla and E. S. Titi, Postprocessing Galerkin methods: The finite-element case, (submitted).
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
- F. Jauberteau, C. Rosier, and R. Temam, A nonlinear Galerkin method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 80 (1990), no. 1-3, 245–260. Spectral and high order methods for partial differential equations (Como, 1989). MR 1067953, DOI 10.1016/0045-7825(90)90028-K
- M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Phys. D 44 (1990), no. 1-2, 38–60. MR 1069671, DOI 10.1016/0167-2789(90)90046-R
- M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations 3 (1991), no. 2, 179–197. MR 1109435, DOI 10.1007/BF01047708
- D. A. Jones, L. G. Margolin and E. S. Titi, On the effectiveness of the approximate inertial manifold —A computational study, Theor. Comp. Fluid Dynam., 7, 1995, 243–260.
- Linda M. Holt, Singularities produced in conormal wave interactions, Trans. Amer. Math. Soc. 347 (1995), no. 1, 289–315. MR 1264146, DOI 10.1090/S0002-9947-1995-1264146-3
- Don A. Jones and Edriss S. Titi, $C^1$ approximations of inertial manifolds for dissipative nonlinear equations, J. Differential Equations 127 (1996), no. 1, 54–86. MR 1387257, DOI 10.1006/jdeq.1996.0061
- Lois Mansfield, On the solution of nonlinear finite element systems, SIAM J. Numer. Anal. 17 (1980), no. 6, 752–765. MR 595441, DOI 10.1137/0717063
- Martine Marion and Roger Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal. 26 (1989), no. 5, 1139–1157. MR 1014878, DOI 10.1137/0726063
- M. Marion and R. Temam, Nonlinear Galerkin methods: the finite elements case, Numer. Math. 57 (1990), no. 3, 205–226. MR 1057121, DOI 10.1007/BF01386407
- Martine Marion and Jinchao Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal. 32 (1995), no. 4, 1170–1184. MR 1342288, DOI 10.1137/0732054
- J. Novo, Postproceso de Métodos espectrales. Ph. D. Thesis, Universidad de Valladolid, 1997.
- Reimund Rautmann, On the convergence rate of nonstationary Navier-Stokes approximations, Approximation methods for Navier-Stokes problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979) Lecture Notes in Math., vol. 771, Springer, Berlin, 1980, pp. 425–449. MR 566012
- Jie Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal. 38 (1990), no. 4, 201–229. MR 1116181, DOI 10.1080/00036819008839963
- Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967, DOI 10.1007/978-1-4684-0313-8
- R. Temam, Attractors for the Navier-Stokes equations: localization and approximation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 629–647. MR 1039488
- R. Temam, Dynamical systems, turbulence and the numerical solution of the Navier-Stokes equations, 11th International Conference on Numerical Methods in Fluid Dynamics (Williamsburg, VA, 1988) Lecture Notes in Phys., vol. 323, Springer, Berlin, 1989, pp. 79–98. MR 1002808, DOI 10.1007/3-540-51048-6_{7}
- Edriss S. Titi, On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. Appl. 149 (1990), no. 2, 540–557. MR 1057693, DOI 10.1016/0022-247X(90)90061-J
- Ronnie Wallace and David M. Sloan, Numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds, SIAM J. Sci. Comput. 16 (1995), no. 5, 1049–1070. MR 1346292, DOI 10.1137/0916060
- Jinchao Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231–237. MR 1257166, DOI 10.1137/0915016
- Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759–1777. MR 1411848, DOI 10.1137/S0036142992232949
Additional Information
- Bosco García-Archilla
- Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
- Email: bosco.garcia@uam.es
- Julia Novo
- Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
- Email: jnovo@mac.mac.cie.uva.es
- Edriss S. Titi
- Affiliation: Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3875, USA
- MR Author ID: 172860
- Email: etiti@math.uci-edu
- Received by editor(s): June 19, 1996
- Received by editor(s) in revised form: February 9, 1998
- Published electronically: February 19, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 893-911
- MSC (1991): Primary 65P25
- DOI: https://doi.org/10.1090/S0025-5718-99-01057-1
- MathSciNet review: 1627785