Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-03T16:16:16.167Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Element Simulations with Adaptively Moving Mesh for the Reaction Diffusion System

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  17 November 2016

Congcong Xie  and
Xianliang Hu
Congcong Xie*
Affiliation:
College of Science, Zhejiang University of Technology, Hangzhou 310023, China
Xianliang Hu*
Affiliation:
School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
*
*Corresponding author. Email addresses:xlhu@zju.edu.cn (X. Hu), ccxie@zjut.edu.cn (C. Xie)
*Corresponding author. Email addresses:xlhu@zju.edu.cn (X. Hu), ccxie@zjut.edu.cn (C. Xie)
Get access

Abstract

A moving mesh method is proposed for solving reaction-diffusion equations. The finite element method is used to solving the partial different equation system, and an efficient numerical scheme is applied to implement mesh moving. In the practical calculations, the moving mesh step and the problem equation solver are performed alternatively. Several numerical examples are presented, including the Gray-Scott, the Activator-Inhibitor and a case with a growing domain. It is illustrated numerically that the moving mesh methods costs much lower, compared with the numerical schemes on a fixed mesh. Even in the case of complex pattern dynamics described by the reaction-diffusion systems, the adapted meshes can capture the details successfully.

Keywords

moving meshfinite elementsreaction-diffusiongrowing domainpattern formation

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N50: Mesh generation and refinement
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 4 , November 2016 , pp. 686 - 704
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, R.A. and Fournier, J.J.F., Sobolev Spaces, Number 140 in Pure and Applied Mathematics, Elsevier Science, 2003.Google Scholar
[2] Ascher, U. M., Ruuth, S. J., and Wetton, B. T. R., Implicit-explicit methods for time-dependent pdes, SIAM J. Numer. Anal., 32 (1997), pp.797–823.Google Scholar
[3] Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, 2nd edition, April 2002.Google Scholar
[4] Ceniceros, H. D. and Hou, T. Y., An efficient dynamically adaptive mesh for potentially singular solutions, J. Comput. Phys., 172 (2001), pp. 609–639.Google Scholar
[5] Ciarlet, P. G., The finite element method for elliptic problems, SIAM: Society for Industrial and Applied Mathematics, 2nd edition, 2002.Google Scholar
[6] Garvie, M. R., Finite-difference schemes for reaction-diffusion equations modeling predatorprey interactions in matlab, Bulletin of Mathematical Biology, 69 (2007), pp. 931–956.CrossRefGoogle ScholarPubMed
[7] Hu, G. and Zegeling, P. A., Simulating finger phenomena in porous media with a moving finite element method, J. Comput. Phys., 230 (2011), pp. 3249–3263.CrossRefGoogle Scholar
[8] Huang, W., Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comp., 3 (1999), pp. 998–1015.Google Scholar
[9] Kassam, A. and Trefethen, L. N., Solving reaction-diffusion equations 10 times faster, Oxford University, Numerical Analysis Group Research Report No.16, 2003.Google Scholar
[10] Kimura, M., Komura, H., Mimura, M., Miroshi, H., Takaishi, T., and Ueyama, D., Adaptive mesh finite element method for pattern dynamics in reaction-diffusion systems, Proceedings of the Czech-Japanese Seminar in Applied Mathematics, 1 (2005), pp. 56–68.Google Scholar
[11] Lee, S. S. and Gaffney, E. A., Aberrant behaviors of reaction diffusion self-organisation models on growing domains in the presence of gene expression time delays, Bulletin of Mathematical Biology, 72 (2010), pp. 2161–2179.Google Scholar
[12] Li, R., Tang, T., and Zhang, P., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170 (2001), pp. 562–588.Google Scholar
[13] Liang, K., Lin, P., Ong, M. T., and Tan, R. C. E., A splitting moving mesh method for reaction-diffusion equations of quenching type, J. Comput. Phys., 215 (2006), pp. 757–777.Google Scholar
[14] Mackenzie, J. A., The efficient generation of simple two-dimensional adaptive grids, SIAM J. Sci. Comput., 19 (1998), pp. 1340–1365.Google Scholar
[15] Madzvamuse, A., Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Comput. Phys., 214 (2006), pp. 239–263.Google Scholar
[16] Mimura, M., Pattern formation in consumer-finite resource reaction-diffusion systems, Publ. RIMS, Kyoto Univ., 40 (2004), pp. 1413–1431.CrossRefGoogle Scholar
[17] Murray, J. D., Mathematical Biology 2nd. ed., Springer-Verlag, 1993.Google Scholar
[18] Page, K. M., Maini, P. K., and Monk, N. A. M., Complex pattern formation in reaction-diffusion systems with spatially varying parameters, Physica D, 202 (2005), pp. 95–115.CrossRefGoogle Scholar
[19] Pearson, J. A., Complex patterns in a simple system, Science, 261 (1993), pp. 189–192.Google Scholar
[20] Tan, Z., Adaptive moving mesh methods for two-dimensional resistive magnetohydrodynamic PDE models, Comput. & Fluids, 36 (2006), pp. 758–771.Google Scholar
[21] Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics), Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006.Google Scholar
[22] Wang, H.-Y., Li, R., and Tang, T., Efficient computation of dendritic growth with r-adaptive finite element methods, J. Comput. Phys., 227 (2008), pp. 5984–6000.Google Scholar
[23] De Zeeuw, P.M., Matrix-dependent prolongations and restrictions in a blackbox multi-grid solver, J. Comput. Appl. Math., 33 (1990), pp. 1–27.CrossRefGoogle Scholar
[24] Zegeling, P. A. and Kok, H. P., Adaptive moving mesh computations for reaction-diffusion systems, J. Comput. Appl. Math., 168 (2004), pp. 519–528.Google Scholar
[25] Zhang, Z. and Tang, T., An adaptive mesh redistribution algorithm for convection-dominated problems, Comm. Pure Appl. Anal., 1 (2002), pp. 341–357.CrossRefGoogle Scholar