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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 |
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. |
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), 457-477.
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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