American Institute of Mathematical Sciences

Advanced
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

Siu-Long Lei 1,

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.
Keywords: Adaptive method, boundary spike., mesh generation, Gierer-Meinhardt system.
Mathematics Subject Classification: Primary: 65Mxx, 14E15; Secondary: 33F0.
Citation: Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete & 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.  doi: 10.1090/S0025-5718-98-00930-2.  Google Scholar

[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.  doi: 10.1137/0732037.  Google Scholar

[3]

M. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics,, J. Comput. Phys., 82 (1989), 64.  doi: 10.1016/0021-9991(89)90035-1.  Google Scholar

[4]

J. Brackbill and J. Saltzman, Adaptive zoning for singular problems in two dimensions,, J. Comput. Phys., 46 (1982), 342.  doi: 10.1016/0021-9991(82)90020-1.  Google Scholar

[5]

M. Del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system,, Commun. Pure Appl. Anal., 1 (2002), 437.  doi: 10.3934/cpaa.2002.1.437.  Google Scholar

[6]

H. Edelsbrunner, "Geometry and Topology for Mesh Generation,", Cambridge University Press, (2001).  doi: 10.1017/CBO9780511530067.  Google Scholar

[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).  doi: 10.1145/1236463.1236465.  Google Scholar

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik (Berlin), 12 (1972), 30.  doi: 10.1007/BF00289234.  Google Scholar

[9]

W. Z. Huang and D. M. Sloan, A simple adaptive grid method in two dimensions,, SIAM J. Sci. Comput., 15 (1994), 776.  doi: 10.1137/0915049.  Google Scholar

[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.  doi: 10.1017/S0956792500004253.  Google Scholar

[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.  doi: 10.1006/jcph.2001.6749.  Google Scholar

[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.  doi: 10.1006/jcph.2002.7002.  Google Scholar

[13]

K. Miller and R. N. Miller, Moving finite elements I,, SIAM J. Numer. Anal., 18 (1981), 1019.  doi: 10.1137/0718070.  Google Scholar

[14]

R. Nicolaides, Direct discretization of planar div-curl problems,, SIAM Numer. Anal., 29 (1992), 32.  doi: 10.1137/0729003.  Google Scholar

[15]

P. O. Persson, "Mesh Generation for Implicit Geometries,", Ph.D thesis, (2005).   Google Scholar

[16]

P. O. Persson and G. Strang, A simple mesh generation in Matlab,, SIAM Rev., 46 (2004), 329.  doi: 10.1137/S0036144503429121.  Google Scholar

[17]

Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system,, Commun. Comput. Phys., 3 (2008), 406.   Google Scholar

[18]

S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation,, J. Math. Biol., 34 (1995), 148.  doi: 10.1007/BF00178771.  Google Scholar

[19]

W. Ren and X. P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions,, J. Comput. Phys., 159 (2000), 246.  doi: 10.1006/jcph.2000.6435.  Google Scholar

[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.  doi: 10.1137/S0036139900375112.  Google Scholar

[21]

J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415.  doi: 10.1007/s00332-001-0380-1.  Google Scholar

show all references

References:
[1]

G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems,, Math. Comput., 67 (1998), 457.  doi: 10.1090/S0025-5718-98-00930-2.  Google Scholar

[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.  doi: 10.1137/0732037.  Google Scholar

[3]

M. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics,, J. Comput. Phys., 82 (1989), 64.  doi: 10.1016/0021-9991(89)90035-1.  Google Scholar

[4]

J. Brackbill and J. Saltzman, Adaptive zoning for singular problems in two dimensions,, J. Comput. Phys., 46 (1982), 342.  doi: 10.1016/0021-9991(82)90020-1.  Google Scholar

[5]

M. Del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system,, Commun. Pure Appl. Anal., 1 (2002), 437.  doi: 10.3934/cpaa.2002.1.437.  Google Scholar

[6]

H. Edelsbrunner, "Geometry and Topology for Mesh Generation,", Cambridge University Press, (2001).  doi: 10.1017/CBO9780511530067.  Google Scholar

[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).  doi: 10.1145/1236463.1236465.  Google Scholar

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetik (Berlin), 12 (1972), 30.  doi: 10.1007/BF00289234.  Google Scholar

[9]

W. Z. Huang and D. M. Sloan, A simple adaptive grid method in two dimensions,, SIAM J. Sci. Comput., 15 (1994), 776.  doi: 10.1137/0915049.  Google Scholar

[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.  doi: 10.1017/S0956792500004253.  Google Scholar

[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.  doi: 10.1006/jcph.2001.6749.  Google Scholar

[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.  doi: 10.1006/jcph.2002.7002.  Google Scholar

[13]

K. Miller and R. N. Miller, Moving finite elements I,, SIAM J. Numer. Anal., 18 (1981), 1019.  doi: 10.1137/0718070.  Google Scholar

[14]

R. Nicolaides, Direct discretization of planar div-curl problems,, SIAM Numer. Anal., 29 (1992), 32.  doi: 10.1137/0729003.  Google Scholar

[15]

P. O. Persson, "Mesh Generation for Implicit Geometries,", Ph.D thesis, (2005).   Google Scholar

[16]

P. O. Persson and G. Strang, A simple mesh generation in Matlab,, SIAM Rev., 46 (2004), 329.  doi: 10.1137/S0036144503429121.  Google Scholar

[17]

Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system,, Commun. Comput. Phys., 3 (2008), 406.   Google Scholar

[18]

S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation,, J. Math. Biol., 34 (1995), 148.  doi: 10.1007/BF00178771.  Google Scholar

[19]

W. Ren and X. P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions,, J. Comput. Phys., 159 (2000), 246.  doi: 10.1006/jcph.2000.6435.  Google Scholar

[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.  doi: 10.1137/S0036139900375112.  Google Scholar

[21]

J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415.  doi: 10.1007/s00332-001-0380-1.  Google Scholar

[1]

Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115
[2]

Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437
[3]

Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363
[4]

Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583
[5]

Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033
[6]

Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885
[7]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
[8]

Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158
[9]

Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
[10]

Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443
[11]

Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521
[12]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002
[13]

Zheng-Ru Zhang, Tao Tang. An adaptive mesh redistribution algorithm for convection-dominated problems. Communications on Pure & Applied Analysis, 2002, 1 (3) : 341-357. doi: 10.3934/cpaa.2002.1.341
[14]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553
[15]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems & Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737
[16]

Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669
[17]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
[18]

Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
[19]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[20]

Aldana M. González Montoro, Ricardo Cao, Christel Faes, Geert Molenberghs, Nelson Espinosa, Javier Cudeiro, Jorge Mariño. Cross nearest-spike interval based method to measure synchrony dynamics. Mathematical Biosciences & Engineering, 2014, 11 (1) : 27-48. doi: 10.3934/mbe.2014.11.27

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences