Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain

Siu-Long Lei

doi:10.3934/dcdsb.2011.15.651

Article Contents

2011, Volume 15, Issue 3: 651-668. Doi: 10.3934/dcdsb.2011.15.651

This issue Previous Article Lipschitz continuous data dependence of sweeping processes in BV spaces Next Article On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model

Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain

Siu-Long Lei^1,

1.
Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau

Received: December 31, 2009

Revised: May 31, 2010

Published: September 2011

Abstract Related Papers Cited by

Abstract

In this paper, an adaptive numerical method is proposed to solve the Gierer-Meinhardt (GM) system on irregular domain. The method works for domains defined by level sets of implicit functions and the generated mesh is of high quality. The method is shown to be effective by comparing with asymptotic result. Boundary spike solutions of the GM system are obtained and studied numerically, including stability of boundary spike and spike motion along the boundary.

Keywords:

Mathematics Subject Classification: Primary: 65Mxx, 14E15; Secondary: 33F05.

Citation:

Related Papers

Cited by

References

[1]	G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems, Math. Comput., 67 (1998), 457-477.doi: 10.1090/S0025-5718-98-00930-2.
[2]	U. M. Ascher, S. J. Ruuth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAMJ. Numer. Anal., 32 (1995), 797-823.doi: 10.1137/0732037.
[3]	M. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82 (1989), 64-84.doi: 10.1016/0021-9991(89)90035-1.
[4]	J. Brackbill and J. Saltzman, Adaptive zoning for singular problems in two dimensions, J. Comput. Phys., 46 (1982), 342-368.doi: 10.1016/0021-9991(82)90020-1.
[5]	M. Del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Commun. Pure Appl. Anal., 1 (2002), 437-456.doi: 10.3934/cpaa.2002.1.437.
[6]	H. Edelsbrunner, "Geometry and Topology for Mesh Generation," Cambridge University Press, 2001.doi: 10.1017/CBO9780511530067.
[7]	N. I. M. Gould, J. A. Scott and Y. Hu, A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations, ACM Trans. Math. Softw., 33(2) (2007), Article 10.doi: 10.1145/1236463.1236465.
[8]	A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39.doi: 10.1007/BF00289234.
[9]	W. Z. Huang and D. M. Sloan, A simple adaptive grid method in two dimensions, SIAM J. Sci. Comput., 15 (1994), 776-797.doi: 10.1137/0915049.
[10]	D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model, Eur. J. of Appl. Math., 11 (2000), 491-514.doi: 10.1017/S0956792500004253.
[11]	R. Li, T. Tang and P. W. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170 (2001), 562-588.doi: 10.1006/jcph.2001.6749.
[12]	R. Li, T. Tang and P. W. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, J. Comput. Phys., 177 (2002), 365-393.doi: 10.1006/jcph.2002.7002.
[13]	K. Miller and R. N. Miller, Moving finite elements I, SIAM J. Numer. Anal., 18 (1981), 1019-1032.doi: 10.1137/0718070.
[14]	R. Nicolaides, Direct discretization of planar div-curl problems, SIAM Numer. Anal., 29 (1992), 32-56.doi: 10.1137/0729003.
[15]	P. O. Persson, "Mesh Generation for Implicit Geometries," Ph.D thesis, MIT, 2005.
[16]	P. O. Persson and G. Strang, A simple mesh generation in Matlab, SIAM Rev., 46 (2004), 329-345.doi: 10.1137/S0036144503429121.
[17]	Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Commun. Comput. Phys., 3 (2008), 406-426.
[18]	S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34 (1995), 148-176.doi: 10.1007/BF00178771.
[19]	W. Ren and X. P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys., 159 (2000), 246-273.doi: 10.1006/jcph.2000.6435.
[20]	M. J. Ward, D. McInerney and P. Houston, The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.doi: 10.1137/S0036139900375112.
[21]	J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458.doi: 10.1007/s00332-001-0380-1.