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Implicit-explicit multistep finite element methods for nonlinear parabolic problems. (English) Zbl 0896.65066
The work focuses on the implicit-explicit multistep finite element method for nonlinear parabolic problems. Particularly, the following aspects are covered: the analysis of a simple one-step semi-explicit scheme of first-order accuracy, general multistep schemes of higher accuracy, the results of three applications: the Kuramoto-Sivashinsky equation and the Cahn-Hilliard equation in one dimension, and a class of reaction-diffusion equations in \(\mathbb{R}^s\), \(s= 2,3\).

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
34A34 Nonlinear ordinary differential equations and systems, general theory
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[1] Georgios Akrivis, High-order finite element methods for the Kuramoto-Sivashinsky equation, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 2, 157 – 183 (English, with English and French summaries). · Zbl 0842.76035
[2] S.M. Allen and J.W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1085-1095.
[3] Michel Crouzeix, Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques, Numer. Math. 35 (1980), no. 3, 257 – 276 (French, with English summary). · Zbl 0419.65057 · doi:10.1007/BF01396412 · doi.org
[4] Michel Crouzeix and Pierre-Arnaud Raviart, Approximation des équations d’évolution linéaires par des méthodes à pas multiples, C. R. Acad. Sci. Paris Sér. A-B 28 (1976), no. 6, Aiv, A367 – A370 (French, with English summary). · Zbl 0361.65064
[5] Colin W. Cryer, A new class of highly-stable methods: \?\(_{0}\)-stable methods, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 153 – 159. · Zbl 0265.65036
[6] Charles M. Elliott and Zheng Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), no. 4, 339 – 357. · Zbl 0624.35048 · doi:10.1007/BF00251803 · doi.org
[7] L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097 – 1123. · Zbl 0801.35045 · doi:10.1002/cpa.3160450903 · doi.org
[8] R. D. Grigorieff and J. Schroll, Über \?(\?)-stabile Verfahren hoher Konsistenzordnung, Computing 20 (1978), no. 4, 343 – 350 (German, with English summary). · Zbl 0395.65043 · doi:10.1007/BF02252382 · doi.org
[9] E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. · Zbl 0729.65051
[10] Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. · Zbl 0558.76051
[11] Marie-Noëlle Le Roux, Semi-discrétisation en temps pour les équations d’évolution paraboliques lorsque l’opérateur dépend du temps, RAIRO Anal. Numér. 13 (1979), no. 2, 119 – 137 (French, with English summary). · Zbl 0413.65066
[12] W.R. McKinney, Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations, Ph.D. thesis, University of Tennessee, Knoxville, 1989.
[13] Basil Nicolaenko and Bruno Scheurer, Remarks on the Kuramoto-Sivashinsky equation, Phys. D 12 (1984), no. 1-3, 391 – 395. · Zbl 0576.35058 · doi:10.1016/0167-2789(84)90543-8 · doi.org
[14] D. T. Papageorgiou, C. Maldarelli, and D. S. Rumschitzki, Nonlinear interfacial stability of core-annular film flows, Phys. Fluids A 2 (1990), no. 3, 340 – 352. · Zbl 0704.76060 · doi:10.1063/1.857784 · doi.org
[15] Giuseppe Savaré, \?(\Theta )-stable approximations of abstract Cauchy problems, Numer. Math. 65 (1993), no. 3, 319 – 335. · Zbl 0791.65077 · doi:10.1007/BF01385755 · doi.org
[16] Larry L. Schumaker, Spline functions: basic theory, John Wiley & Sons, Inc., New York, 1981. Pure and Applied Mathematics; A Wiley-Interscience Publication. · Zbl 0449.41004
[17] Eitan Tadmor, The well-posedness of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 17 (1986), no. 4, 884 – 893. · Zbl 0606.35073 · doi:10.1137/0517063 · doi.org
[18] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. · Zbl 0662.35001
[19] Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. · Zbl 0884.65097
[20] Miloš Zlámal, Finite element multistep discretizations of parabolic boundary value problems, Math. Comp. 29 (1975), 350 – 359.
[21] Miloš Zlámal, Finite element methods for nonlinear parabolic equations, RAIRO Anal. Numér. 11 (1977), no. 1, 93 – 107, 113 (English, with French summary).
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