Finite element analysis of chemotaxis-growth model with indirect attractant production and logistic source

References

• R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Elsevier, 2003.
   Google Scholar
• G.A. Al-Juaifri and A.J. Harfash, Finite element analysis of nonlinear reaction–diffusion system of fitzhugh–nagumo type with robin boundary conditions, Math. Comput. Simul. 203 (2023), pp. 486–517.
   Web of Science ®Google Scholar
• J.W. Barrett and J.F. Blowey, Finite element approximation of a nonlinear cross-diffusion population model, Numer. Math. 98(2) (2004), pp. 195–221.
   Web of Science ®Google Scholar
• J.W. Barrett, H. Garcke, and R. Nürnberg, Finite element approximation of surfactant spreading on a thin film, SIAM. J. Numer. Anal. 41(4) (2003), pp. 1427–1464.
   Web of Science ®Google Scholar
• J.W. Barrett and R. Nürnberg, Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation, Interfaces Free Boun. 4(3) (2002), pp. 277–307.
   Web of Science ®Google Scholar
• J.W. Barrett and R. Nürnberg, Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der waals forces, IMA J. Numer. Anal. 24(2) (2004), pp. 323–363.
   Web of Science ®Google Scholar
• J.W. Barrett, R. Nürnberg, and M. RE Warner, Finite element approximation of soluble surfactant spreading on a thin film, SIAM. J. Numer. Anal. 44(3) (2006), pp. 1218–1247.
   Web of Science ®Google Scholar
• J.W. Barrett, C. Schwab, and E. Süli, Existence of global weak solutions for some polymeric flow models, Math. Models Methods Appl. Sci. 15(06) (2005), pp. 939–983.
   Web of Science ®Google Scholar
• N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of keller–segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25(09) (2015), pp. 1663–1763.
   Web of Science ®Google Scholar
• T. Cazenave, Semilinear Schrodinger Equations, Vol. 10, American Mathematical Society, Providence, RI, 2003.
   Google Scholar
• A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math. 111(2) (2008), pp. 169.
   Web of Science ®Google Scholar
• P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002.
   Google Scholar
• J.F. Ciavaldini, Analyse numerique d'un problème de stefan à deux phases par une methode d'éléments finis, SIAM. J. Numer. Anal. 12(3) (1975), pp. 464–487.
   Web of Science ®Google Scholar
• Y. Epshteyn, Upwind-difference potentials method for patlak-keller-segel chemotaxis model, J. Sci. Comput. 53(3) (2012), pp. 689–713.
   Web of Science ®Google Scholar
• Y. Epshteyn and A. Izmirlioglu, Fully discrete analysis of a discontinuous finite element method for the keller-segel chemotaxis model, J. Sci. Comput. 40(1) (2009), pp. 211–256.
   Web of Science ®Google Scholar
• Y. Epshteyn and A. Kurganov, New interior penalty discontinuous galerkin methods for the keller–segel chemotaxis model, SIAM. J. Numer. Anal. 47(1) (2009), pp. 386–408.
   Web of Science ®Google Scholar
• F. Filbet, A finite volume scheme for the patlak–keller–segel chemotaxis model, Numer. Math. 104(4) (2006), pp. 457–488.
   Web of Science ®Google Scholar
• H. Fu and H. Rui, A priori and a posteriori error estimates for the method of lumped masses for parabolic optimal control problems, Int. J. Comput. Math. 88(13) (2011), pp. 2798–2823.
   Web of Science ®Google Scholar
• G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math. 87(1) (2000), pp. 113–152.
   Web of Science ®Google Scholar
• L. Guo and F. Huang, Convergence and quasi-optimality of an adaptive finite element method for semilinear elliptic problems on L2 errors, Int. J. Comput. Math. 97(11) (2020), pp. 2337–2354.
   Web of Science ®Google Scholar
• M.H. Hashim and A.J. Harfash, Finite element analysis of a keller–segel model with additional cross-diffusion and logistic source. part I: space convergence, Comput. Math. with Appl. 89(1) (2021), pp. 44–56.
   Web of Science ®Google Scholar
• M.H. Hashim and A.J. Harfash, Finite element analysis of a keller-segel model with additional cross-diffusion and logistic source. part II: time convergence and numerical simulation, Comput. Math. with Appl. 109(1) (2022), pp. 216–234.
   Web of Science ®Google Scholar
• M.H. Hashim and A.J. Harfash, Finite element analysis of attraction-repulsion chemotaxis system. part I: space convergence, Commun, Appl. Math. Comput. 4(3) (2022), pp. 1011–1056.
   Web of Science ®Google Scholar
• M.H. Hashim and A.J. Harfash, Finite element analysis of attraction-repulsion chemotaxis system. part II: time convergence, error analysis and numerical results, Commun. Appl. Math. Comput. 4(3) (2022), pp. 1057–1104.
   Web of Science ®Google Scholar
• S.M. Hassan and A.J. Harfash, Finite element analysis of a two-species chemotaxis system with two chemicals, Appl. Numer. Math. 182 (2022), pp. 148–175.
   Web of Science ®Google Scholar
• S.M. Hassan and A.J Harfash, Finite element approximation of a keller–segel model with additional self-and cross-diffusion terms and a logistic source, Commun. Nonlinear Sci. Numer. Simul. 104 (2022), pp. 106063.
   Web of Science ®Google Scholar
• M.A. Herrero and J.J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Della Scuola Norm. Superiore Di Pisa-Classe Di Sci. 24(4) (1997), pp. 633–683.
   Google Scholar
• T. Hillen and K.J. Painter, A user's guide to pde models for chemotaxis, J. Math. Biol. 58(1–2) (2009), pp. 183.
   PubMed Web of Science ®Google Scholar
• T. Hou, L∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems, Int. J. Comput. Math. 89(16) (2012), pp. 2224–2239.
   Web of Science ®Google Scholar
• E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26(3) (1970), pp. 399–415.
   PubMed Web of Science ®Google Scholar
• H.-C. Lee, Weighted least-squares finite element methods for the linearized navier–stokes equations, Int. J. Comput. Math. 91(9) (2014), pp. 1964–1985.
   Web of Science ®Google Scholar
• Y. Li, J. Han, Y. Yang, and H. Bi, The a priori and a posteriori error estimates of crouzeix–raviart element for the fluid–solid vibration problem, Int. J. Comput. Math. 99(10) (2022), pp. 1968–1988.
   Google Scholar
• J. Manimaran and L. Shangerganesh, Error estimates for galerkin finite element approximations of time-fractional nonlocal diffusion equation, Int. J. Comput. Math. 98(7) (2021), pp. 1365–1384.
   Web of Science ®Google Scholar
• A. Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixed finite elements, ESAIM: Math. Model Numer. Anal. 37(4) (2003), pp. 617–630.
   Web of Science ®Google Scholar
• R.H. Nochetto, Finite element methods for parabolic free boundary problems, in Advances in Numerical Analysis: Nonlinear Partial Differential Equations and Dynamical Systems, Vol. 1, W. Light, ed., Oxford University Press, New York, 1991, pp. 34–95.
   Google Scholar
• R.H. Nochetto and C. Verdi, Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math. 74(1) (1996), pp. 105–136.
   Web of Science ®Google Scholar
• J.C Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, 28, Cambridge University Press, 2001.
   Google Scholar
• N. Saito, Conservative upwind finite-element method for a simplified keller–segel system modelling chemotaxis, IMA J. Numer. Anal. 27(2) (2007), pp. 332–365.
   Web of Science ®Google Scholar
• W.G. Strang and J. George, An Analysis of the Finite Element Method, Prentice-Hall, Upper Saddle River, 1973.
   Google Scholar
• R. Strehl, A. Sokolov, D. Kuzmin, and S. Turek, A flux-corrected finite element method for chemotaxis problems, Comput. Methods Appl. Math. 10(2) (2010), pp. 219–232.
   Google Scholar
• S. Strohm, R.C. Tyson, and J.A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol. 75(10) (2013), pp. 1778–1797.
   PubMed Web of Science ®Google Scholar
• R. Tyson, S.R. Lubkin, and J.D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol. 38(4) (1999), pp. 359–375.
   PubMed Web of Science ®Google Scholar
• R. Tyson, L.G. Stern, and R.J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol. 41(5) (2000), pp. 455–475.
   PubMed Web of Science ®Google Scholar
• D.E. Woodward, R. Tyson, M.R. Myerscough, J.D. Murray, E.O. Budrene, and H.C. Berg, Spatio-temporal patterns generated by salmonella typhimurium, Biophys. J. 68(5) (1995), pp. 2181–2189.
   PubMed Web of Science ®Google Scholar
• C. Yang, Convergence of linearized backward euler–galerkin finite element methods for the time-dependent ginzburg–landau equations with temporal gauge, Int. J. Comput. Math. 91(7) (2014), pp. 1507–1515.
   Web of Science ®Google Scholar
• L. Zhu and Z. Zhou, Convergence and quasi-optimality of an adaptive continuous interior multi-penalty finite element method, Int. J. Comput. Math. 97(9) (2020), pp. 1884–1907.
   Web of Science ®Google Scholar