Skip to main content
SpringerLink
Log in
Book cover

Numerical Mathematics and Advanced Applications 2011 pp 71–80Cite as

Adaptive Finite Elements for Semilinear Reaction-Diffusion Systems on Growing Domains

  • Conference paper
  • First Online:

Abstract

We propose an adaptive finite element method to approximate the solutions to reaction-diffusion systems on time-dependent domains and surfaces. We derive a computable error estimator that provides an upper bound for the error in the semidiscrete (space) scheme. We reconcile our theoretical results with benchmark computations.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The research of CV has been supported by the EPSRC, Grant EP/G010404. This work (AM) is partly supported by the following grants: EPSRC (EP/H020349/1), the LMS grant (R4P2), and the British Council through its UK-US New Partnership Fund (PMI2).

References

  1. A. Madzvamuse. A Numerical Approach to the Study of Spatial Pattern Formation. DPhil thesis, University of Oxford, 2000.

    Google Scholar 

  2. C. Venkataraman. Reaction-diffusion systems on evolving domains with applications to the theory of biological pattern formation. DPhil thesis, University of Sussex, June 2011. URL http://sro.sussex.ac.uk/6908/.

  3. O. Lakkis, A. Madzvamuse, and C. Venkataraman. Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains. Technical report, March 2012. URL http://adsabs.harvard.edu/abs/2011arXiv1111.5052L.

  4. O. Kruger, M. Picasso, and JF Scheid. A posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification. Computer Methods in Applied Mechanics and Engineering, 192(5–6):535–558, 2003.

    Google Scholar 

  5. Chandrasekhar Venkataraman, Omar Lakkis, and Anotida Madzvamuse. Global existence for semilinear reaction–diffusion systems on evolving domains. Journal of Mathematical Biology, 64:41–67, 2012. ISSN 0303-6812. URL http://dx.doi.org/10.1007/s00285-011-0404-x.

  6. J. Medina, M. Picasso, and J. Rappaz. Error estimates and adaptive finite elements for nonlinear diffusion-convection problems. Mathematical Models and Methods in Applied Sciences, 6(5):689–712, 1996. ISSN 0218-2025.

    Google Scholar 

  7. A. Madzvamuse. A modified backward euler scheme for advection-reaction-diffusion systems. Mathematical Modeling of Biological Systems, Volume I, pages 183–189, 2007.

    Google Scholar 

  8. A. Schmidt and K.G. Siebert. Design of adaptive finite element software: The finite element toolbox ALBERTA. Springer Verlag, 2005.

    Google Scholar 

  9. R. Lefever and I. Prigogine. Symmetry-breaking instabilities in dissipative systems II. J. chem. Phys, 48:1695–1700, 1968.

    Article  Google Scholar 

  10. C. Venkataraman, T. Sekimura, E.A. Gaffney, P.K. Maini, and A. Madzvamuse. Modeling parr-mark pattern formation during the early development of amago trout. Phys. Rev. E, 84:041923, Oct 2011. doi: 10.1103/PhysRevE.84.041923.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK

    C. Venkataraman

  2. Department of Mathematics, University of Sussex, Pev III, Brighton, BN1 9QH, UK

    O. Lakkis & A. Madzvamuse

Authors
  1. C. Venkataraman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Lakkis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Madzvamuse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Venkataraman .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Venkataraman, C., Lakkis, O., Madzvamuse, A. (2013). Adaptive Finite Elements for Semilinear Reaction-Diffusion Systems on Growing Domains. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation