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Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme. (English) Zbl 1172.92311
Summary: We continue our systematic study of the dimension estimate of the global attractors for chemotaxis-growth systems and their finite-element approximations [see Discrete Contin. Dyn. Syst. 2007, Suppl., 334--343 (2007; Zbl 1163.37396)]. Utilizing a conservative upwind finite-element scheme we managed significantly to improve dimension estimates with respect to the chemotactic parameter.

92C17Cell movement (chemotaxis, etc.)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
37N25Dynamical systems in biology
35Q80Applications of PDE in areas other than physics (MSC2000)
Full Text: DOI
[1] Aida, M.; Tsujikawa, T.; Efendiev, M.; Yagi, A.; Mimura, M.: Lower estimate of attractor dimension for chemotaxis growth system, J. lond. Math. soc. 74, 453-474 (2006) · Zbl 1125.37056 · doi:10.1112/S0024610706023015
[2] Aida, M.; Yagi, A.: Global attractor for approximate system of chemotaxis and growth, Dyn. contin. Discrete impuls. Syst. ser. A 10, 309-315 (2003) · Zbl 1038.34053
[3] Baba, K.; Tabata, M.: On a conservative upwind finite element scheme for convective diffusion equations, RAIRO anal. Numér. 15, 3-25 (1981) · Zbl 0466.76090
[4] Babin, A. V.; Vishik, M. I.: Attractors of evolution equations, (1989) · Zbl 0804.58003
[5] Bellman, R.: Introduction to matrix analysis, (1997) · Zbl 0872.15003
[6] Ciarlet, P. G.: The finite element method for elliptic problems, (1978) · Zbl 0383.65058
[7] Efendiev, M.; Nakaguchi, E.: Upper and lower estimate of dimension of the global attractor for the chemotaxis -- growth system I, Adv. math. Sci. appl. 16, 569-579 (2006) · Zbl 1130.37402
[8] Efendiev, M.; Nakaguchi, E.: Upper and lower estimate of dimension of the global attractor for the chemotaxis -- growth system II: Two-dimensional case, Adv. math. Sci. appl. 16, 581-590 (2006) · Zbl 1130.37403
[9] Efendiev, M.; Nakaguchi, E.; Wendland, W. L.: Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis -- growth system, Discrete contin. Dyn. syst. 2007, No. Suppl., 334-343 (2007) · Zbl 1163.37396 · http://www.aimsciences.org/journals/redirecting.jsp?paperID=2816
[10] Fujii, H.: Some remarks on finite element analysis of time-dependent field problems, Theory and practice in finite element structural analysis, 91-106 (1973) · Zbl 0373.65047
[11] Fujita, H.; Saito, N.; Suzuki, T.: Operator theory and numerical methods, (2001) · Zbl 0976.65098
[12] Mimura, M.; Tsujikawa, T.: Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A 230, 499-543 (1996)
[13] Murray, J. D.: Mathematical biology, (2002) · Zbl 1006.92001
[14] Nakaguchi, E.; Efendiev, M.: On a new dimension estimate of the global attractor for chemotaxis -- growth systems, Osaka J. Math. 45, 273-281 (2008) · Zbl 1160.37034 · euclid:ojm/1216151100
[15] Nakaguchi, E.; Yagi, A.: Fully discrete approximations by Galerkin Runge -- Kutta method for quasilinear parabolic systems, Hokkaido math. J. 31, 385-429 (2002) · Zbl 1011.65067
[16] Saito, N.: A holomorphic semigroup approach to the lumped mass finite element method, J. comput. Appl. math. 169, 71-85 (2004) · Zbl 1058.65108 · doi:10.1016/j.cam.2003.11.003
[17] Saito, N.: Conservative upwind finite-element method for a simplified Keller -- segel system modelling chemotaxis, IMA J. Numer. anal. 27, 332-365 (2007) · Zbl 1119.65094 · doi:10.1093/imanum/drl018
[18] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, (1997) · Zbl 0871.35001
[19] Thomée, V.: Galerkin finite element methods for parabolic problems, (1997) · Zbl 0884.65097
[20] Triebel, H.: Interpolation theory, function spaces, differential operators, (1978) · Zbl 0387.46033