# American Institute of Mathematical Sciences

December  2011, 15(3): 651-668. doi: 10.3934/dcdsb.2011.15.651

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

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

Received  January 2010 Revised  June 2010 Published  February 2011

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.
Citation: Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651
##### 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.

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##### 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.

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