Abstract
This paper formulates and analyzes fully discrete schemes for the two-dimensional Keller-Segel chemotaxis model. The spatial discretization of the model is based on the discontinuous Galerkin methods and the temporal discretization is based either on Forward Euler or the second order explicit total variation diminishing (TVD) Runge-Kutta methods. We consider Cartesian grids and prove fully discrete error estimates for the proposed methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adler, J.: Chemotaxis in bacteria. Ann. Rev. Biochem. 44, 341–356 (1975)
Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton (1965)
Arnold, D.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)
Babuška, I., Suri, M.: The hp version of the finite element method with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér. 21, 199–238 (1987)
Babuška, I., Suri, M.: The optimal convergence rates of the p-version of the finite element method. SIAM J. Numer. Anal. 24, 750–776 (1987)
Bonner, J.: The Cellular Slime Molds, 2nd edn. Princeton University Press, Princeton (1967)
Brenner, S.: Poincaré-Friedrichs inequalities for piecewise h 1 functions. SIAM J. Numer. Anal. 41, 306–324 (2003)
Budrene, E., Berg, H.: Complex patterns formed by motile cells of Escherichia coli. Nature 349, 630–633 (1991)
Budrene, E., Berg, H.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376, 49–53 (1995)
Chertock, A., Kurganov, A.: A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205 (2008)
Childress, S., Percus, J.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)
Cockburn, B., Shu, C.-W.: , The local discontinuous Galerkin method for convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.): First International Symposium on Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)
Cockburn, B., Kanschat, G., Schotzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31, 61–73 (2007)
Cohen, M., Robertson, A.: Wave propagation in the early stages of aggregation of cellular slime molds. J. Theor. Biol. 31, 101–118 (1971)
Douglas, J., Dupont, T.: Lecture Notes in Physics, vol. 58. Springer, Berlin (1976). Chap. Interior penalty procedures for elliptic and parabolic Galerkin methods
Epshteyn, Y.: Discontinuous Galerkin methods for the chemotaxis and haptotaxis models. J. Comput. Appl. Math. 224, 168–181 (2008)
Epshteyn, Y., Izmirlioglu, A.: Fully discrete analysis of a discontinuous finite element method for the Keller-Segel Chemotaxis model. CNA Report 08-CNA-026. http://www.andrew.cmu.edu/user/rina10/chemotdiscrevcna.pdf (2008)
Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. SIAM J. Numer. Anal. 47, 386–408 (2008). CNA Report. http://www.math.cmu.edu/cna/pub2007.html
Epshteyn, Y., Rivière, B.: On the solution of incompressible two-phase flow by a p-version discontinuous Galerkin method. Commun. Numer. Methods Eng. 22, 741–751 (2006)
Epshteyn, Y., Rivière, B.: Convergence of high order methods for miscible displacement. Int. J. Numer. Anal. Model. 5(Supp), 47–63 (2008)
Filbet, F.: A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104, 457–488 (2006)
Girault, V., Rivière, B., Wheeler, M.: A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comput. 74, 53–84 (2005)
Herrero, M., Velázquez, J.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. 24, 633–683 (1997)
Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences i. Jahresber. Dtsch. Math.-Ver. 105, 103–165 (2003)
Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ii. Jahresber. Dtsch. Math.-Ver. 106, 51–69 (2004)
Keller, E., Segel, L.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Keller, E., Segel, L.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
Keller, E., Segel, L.: Traveling bands of chemotactic bacteria: A theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)
Kurganov, A., Lin, C.-T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2, 141–163 (2007)
Kurganov, A., Petrova, G.: Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws. Numer. Methods Partial Differ. Equ. 21, 536–552 (2005)
Kurganov, A., Noelle, S., Petrova, G.: Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)
Marrocco, A.: 2d simulation of chemotaxis bacteria aggregation. Math. Model. Numer. Anal. 37, 617–630 (2003)
Nanjundiah, V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)
Patlak, C.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)
Prescott, L., Harley, J., Klein, D.: Microbiology, 3rd edn. Brown, Chicago (1996)
Rivière, B., Wheeler, M., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001)
Schwab, C.: p- and hp-Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, London (1998)
Sun, S., Wheeler, M.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43, 195–219 (2005)
Tyson, R., Lubkin, S., Murray, J.: A minimal mechanism for bacterial pattern formation. Proc. R. Soc. Lond. B 266, 299–304 (1999)
Tyson, R., Stern, L., LeVeque, R.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)
Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641–666 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Epshteyn, Y., Izmirlioglu, A. Fully Discrete Analysis of a Discontinuous Finite Element Method for the Keller-Segel Chemotaxis Model. J Sci Comput 40, 211–256 (2009). https://doi.org/10.1007/s10915-009-9281-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-009-9281-5