A linear discrete scheme for the Ginzburg–Landau equation

References

• Temam , R. 1988 . Infinite Dimensional Dynamical Systems in Mechanics and Physics , New York : Springer .
   Google Scholar
• Ghidaglia , J. R. and Temam , R. 1990 . “ Lower bound on the dimension of the attractor for the Navier–Stokes equations in space 3 ” . In Mechanics, Analysis and Geometry: 200 Years After Lagrange , Edited by: Francariglia , M. Amsterdam : Elsevier .
   Google Scholar
• Hale , J. K. 1988 . Asymptotic behavior of dissipative systems , Vol. 25 , Providence, RI : American Mathematical Society . Mathematical Surveys and Monographs
   Google Scholar
• Eden , A. , Foias , C. , Nicolaenko , B. and Temam , R. 1994 . Exponential Attractors for Dissipative Evolution Equations , New York : Wiley . (Research in Applied Mathematics 37)
   Google Scholar
• Bu , C. 1994 . On the Cauchy problem for the 1+2 complex Ginzburg–Landau equation . Journal of the Australian Mathematics Society , 6 : 313 – 324 . Ser. B
   Google Scholar
• Gao , N. , Lin , G. and Duan , J. 2000 . Asymptotics for the generalized two-dimensional Ginzburg–Landau equation . Journal of Mathematical Analysis and Applications , 247 : 198 – 216 .
   Google Scholar
• Li , Y. and Guo , B. 2000 . Global existence of solutions to the derivative 2D Ginzburg–Landau equation . Journal of Mathematical Analysis and Applications , 247 : 412 – 432 .
   Google Scholar
• Ghidalia , J. M. and Héron , B. 1987 . Dimension of the attractor associated to the Ginzburg–Landau equation . Physica , 18 : 282 – 304 . D
   Google Scholar
• Levermore , C. D. and Oliver , M. 1996 . “ The complex Ginzburg–Landau equation as a model problem ” . In Dynamical Systems and Probabilistic Methods for Nonlinear Waves Vol. 31 , 141 – 189 . (Lectures in Applied Mathematics)
   Google Scholar
• Lü , S. and Lu , Q. 2006 . Asymptotic behavior of three-dimensional Ginzburg–Landau type equation . Dynamics of Continuous, Discrete and Impulsive Systems , 13 : 209 – 220 . Series A
   Google Scholar
• Hale , J. K. , Lin , X. B. and Raugel , G. 1988 . Upper semi-continuity of attractors for approximations of semi-groups and partial differential equations . Mathematics of Computation , 50 : 89 – 123 .
   Web of Science ®Google Scholar
• Elliott , C. M. and Larsson , S. 1992 . Error estimates with smooth and non-smooth data for a finite element method for the Cahn–Hilliard equation . Mathematics of Computation , 58 : 603 – 630 .
   Web of Science ®Google Scholar
• Lü , S. 2000 . The dynamical behavior of the Ginzburg–Landau equation and its Fourier spectral approximation . Numerical Mathematics , 22 : 1 – 9 . (in Chinese)
   Google Scholar
• Lü , S. and Lu , Q. 2007 . Fourier spectral approximation to long-time behavior of three dimensional Ginzburg–Landau type equation . Advances in Computational Mathematics , 27 : 293 – 318 .
   Google Scholar
• Lü , S. and Lu , Q. 2006 . Fourier spectral approximation to long-time behavior of dissipative generalized KdV–Burgers equations . SIAM Journal of Numerical Analysis , 44 : 561 – 585 .
   Google Scholar
• Lü , S. , Lu , Q. and Twizell , E. H. 2007 . Fourier spectral approximation to long-time behavior of the derivative three dimensional Ginzburg–Landau equation . Journal of Computational and Applied Mathematics , 198 : 167 – 186 .
   Web of Science ®Google Scholar
• Marion , R. 1988 . Nonlinear Galerkin methods . SIAM Journal of Numerical Analysis , 26 : 1139 – 1157 .
   Google Scholar
• Marion , M. and Temam , R. 1990 . Nonlinear Galerkin methods: the finite elements case . Numerical Mathematics , 57 : 205 – 226 .
   Web of Science ®Google Scholar
• Chen , M. and Temam , R. 1993 . Nonlinear Galerkin methods in the finite difference case and wavelet-like incremental unknowns . Numerical Mathematics , 64 : 271 – 294 .
   Web of Science ®Google Scholar
• Shen , J. 1990 . Long-time stability and convergence for fully discrete nonlinear Galerkin methods . Applicable Analysis , 38 : 201 – 209 .
   Google Scholar
• Canuto , C. and Qarteroni , A. 1982 . Approximation results for orthogonal polynomials in Sobolev spaces . Mathematics of Computation , 38 : 201 – 229 .
   Google Scholar
• Mazja , V. G. 1985 . Sobolev Space , New York : Springer .
   Google Scholar