A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation
J. Zhu; Y.-T. Zhang; S. A. Newman; M. S. Alber
Mathematical Modelling of Natural Phenomena (2009)
- Volume: 4, Issue: 4, page 131-148
- ISSN: 0973-5348
Access Full Article
topAbstract
topHow to cite
topZhu, J., et al. "A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation." Mathematical Modelling of Natural Phenomena 4.4 (2009): 131-148. <http://eudml.org/doc/222349>.
@article{Zhu2009,
abstract = {
Skeletal patterning in the vertebrate limb,
i.e., the spatiotemporal regulation of cartilage differentiation
(chondrogenesis) during embryogenesis and regeneration, is one
of the best studied examples of a multicellular developmental process.
Recently [Alber et al., The morphostatic limit for a model of
skeletal pattern formation in the vertebrate limb, Bulletin of
Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation
reaction-diffusion system was developed to describe the interaction of two of
the key morphogens: the activator and an activator-dependent inhibitor of
precartilage condensation formation. A discontinuous Galerkin (DG)
finite element method was applied to solve this nonlinear system on complex
domains to study the effects of domain geometry on the pattern generated [Zhu et
al., Application of Discontinuous Galerkin Methods for reaction-diffusion
systems in developmental biology, Journal of Scientific Computing, 2009, v40,
pp. 391-418]. In this paper, we extend these previous results and develop a DG
finite element model in a moving and deforming domain for skeletal pattern
formation in the vertebrate limb. Simulations reflect the actual dynamics of
limb development and indicate the important role played by the geometry
of the undifferentiated apical zone.
},
author = {Zhu, J., Zhang, Y.-T., Newman, S. A., Alber, M. S.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {discontinuous Galerkin finite element methods; reaction-diffusion
equations; operator splitting; triangular meshes; moving domain; complex
geometry; limb development; reaction-diffusion equations; complex geometry},
language = {eng},
month = {7},
number = {4},
pages = {131-148},
publisher = {EDP Sciences},
title = {A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation},
url = {http://eudml.org/doc/222349},
volume = {4},
year = {2009},
}
TY - JOUR
AU - Zhu, J.
AU - Zhang, Y.-T.
AU - Newman, S. A.
AU - Alber, M. S.
TI - A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/7//
PB - EDP Sciences
VL - 4
IS - 4
SP - 131
EP - 148
AB -
Skeletal patterning in the vertebrate limb,
i.e., the spatiotemporal regulation of cartilage differentiation
(chondrogenesis) during embryogenesis and regeneration, is one
of the best studied examples of a multicellular developmental process.
Recently [Alber et al., The morphostatic limit for a model of
skeletal pattern formation in the vertebrate limb, Bulletin of
Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation
reaction-diffusion system was developed to describe the interaction of two of
the key morphogens: the activator and an activator-dependent inhibitor of
precartilage condensation formation. A discontinuous Galerkin (DG)
finite element method was applied to solve this nonlinear system on complex
domains to study the effects of domain geometry on the pattern generated [Zhu et
al., Application of Discontinuous Galerkin Methods for reaction-diffusion
systems in developmental biology, Journal of Scientific Computing, 2009, v40,
pp. 391-418]. In this paper, we extend these previous results and develop a DG
finite element model in a moving and deforming domain for skeletal pattern
formation in the vertebrate limb. Simulations reflect the actual dynamics of
limb development and indicate the important role played by the geometry
of the undifferentiated apical zone.
LA - eng
KW - discontinuous Galerkin finite element methods; reaction-diffusion
equations; operator splitting; triangular meshes; moving domain; complex
geometry; limb development; reaction-diffusion equations; complex geometry
UR - http://eudml.org/doc/222349
ER -
References
top- M. Alber, H.G.E. Hentschel, B. Kazmierczak, S.A. Newman. Existence of solutions to a new model of biological pattern formation. J. Math. Anal. Appl., 308 (2005), No. 1, 175–194.
- M. Alber, T. Glimm, H.G.E. Hentschel, B. Kazmierczak, Y.-T. Zhang, J. Zhu, S.A. Newman. The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb. Bulletin of Mathematical Biology, 70 (2008), No. 2, 460–483.
- Y. Cheng, C.-W. Shu. A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Mathematics of Computation, 77 (2008), No. 262, 699–730.
- B. Cockburn, G. Karniadakis, C.-W. Shu. The development of discontinuous Galerkin methods, in Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. Karniadakis, and C.-W. Shu, Editors. Lecture Notes in Computational Science and Engineering, 11 (2000), Springer, 3–50.
- B. Cockburn, C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16 (2001), No. 3, 173–261.
- B. Cockburn, C.-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numererical Analysis, 35 (1998), No. 6, 2440–2463.
- H.G.E. Hentschel, T. Glimm, J.A. Glazier, S.A. Newman. Dynamical mechanisms for skeletal pattern formation in the vertebrate limb. Proc. R. Soc. B, 271 (2004), No. 1549, 1713–1722.
- W. Hundsdorfer. Trapezoidal and midpoint splittings for initial-boundary value problems. Mathematics of Computation, 67 (1998), No. 223, 1047–1062.
- P.K. Kundu. Fluid Mechanics. Academic Press, Inc, London, 1990.
- D. Levy, C.-W. Shu, J. Yan. Local discontinuous Galerkin methods for nonlinear dispersive equations. Journal of Computational Physics, 196 (2004), No. 2, 751–772.
- A. Madzvamuse, A.J. Wathen, P.K. Maini. A moving grid finite element method applied to a model biological pattern generator. Journal of Computational Physics, 190 (2003), No. 2, 478–500.
- A. Madzvamuse, P.K. Maini, A.J. Wathen. A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J. Sci. Comput., 24 (2005), No. 2, 247–262.
- A. Madzvamuse. Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. Journal of Computational Physics, 214 (2006), No. 1, 239–263.
- C.E. Nelson, B.A. Morgan, A.C. Burke, E. Laufer, E. DiMambro, L.C. Murtaugh, E. Gonzales, L. Tessarollo, L.F. Parada, C. Tabin. Analysis of Hox gene expression in the chick limb bud. Development, 122 (1996), No. 5, 1449–1466.
- S.A. Newman, G.B. Müller. Origination and innovation in the vertebrate limb skeleton: an epigenetic perspective. J. Exp. Zoolog. B Mol. Dev. Evol. 304 (2005), No. 6, 593–609.
- S.A. Newman, R. Bhat. Activator-inhibitor dynamics of vertebrate limb pattern formation. Birth Defects Res C Embryo Today, 81 (2007), No. 4, 305–319.
- S.A. Newman, S. Christley, T. Glimm, H.G.E. Hentschel, B. Kazmierczak, Y.-T. Zhang, J. Zhu, M. Alber. Multiscale models for vertebrate limb development. Curr. Top. Dev. Biol., 81 (2008), 311–340.
- M.A. Ros, G.E. Lyons, S. Mackem, J.F. Fallon. Recombinant limbs as a model to study homeobox gene regulation during limb development. Dev. Biol., 166 (1994), No. 1, 59–72.
- G. Strang. On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 8 (1968), No. 3, 506–517.
- D. Summerbell. A descriptive study of the rate of elongation and differentiation of the skeleton of the developing chick wing. J. Embryol. Exp. Morphol., 35 (1976), No. 2, 241–260.
- T. Svingen, K.F. Tonissen. Hox transcription factors and their elusive mammalian gene targets. Heredity, 97 (2006), No. 2, 88–96.
- C. Tickle. Patterning systems - from one end of the limb to the other. Dev. Cell, 4 (2003), No. 4, 449–458.
- Y. Xu, C.-W. Shu. Local discontinuous Galerkin methods for three classes of nonlinear wave equations. Journal of Computational Mathematics, 22 (2004), No. 2, 250–274.
- Y. Xu, C.-W. Shu. Local discontinuous Galerkin methods for nonlinear Schrodinger equations. Journal of Computational Physics, 205 (2005), No. 1, 72–97.
- Y. Xu, C.-W. Shu. Local discontinuous Galerkin methods for two classes of two dimensional nonlinear wave equations. Physica D, 208 (2005), No. 1-2, 21–58.
- Y. Xu, C.-W. Shu. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Computer Methods in Applied Mechanics and Engineering, 195 (2006), No. 25-28, 3430–3447.
- J. Yan, C.-W. Shu. A local discontinuous Galerkin method for KdV type equations. SIAM Journal on Numerical Analysis, 40 (2002), No. 2, 769–791.
- J. Yan, C.-W. Shu. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. Journal of Scientific Computing, 17 (2002), No. 1-4, 27–47.
- J. Zhu, Y.-T. Zhang, S.A. Newman, M. Alber. Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology. Journal of Scientific Computing, 40 (2009), No. 1-3, 391–418.
- E. Zwilling. Development of fragmented and of dissociated limb bud mesoderm. Dev. Biol., 9 (1964), No. 1, 20–37.
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.