The surface finite element method for pattern formation on evolving biological surfaces

• Aragón JL, Barrio RA, Varea C (1999) Turing patterns on a sphere. Phys Rev E 60: 4588–4592

  Article  Google Scholar 

• Barreira R (2009) Numerical solution of non-linear partial differential equations on triangulated surfaces, D.Phil Thesis, University of Sussex

• Barrett JW, Garcke H, Nürnberg (2008) Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J Sci Comput 31: 225–253

  Article  MathSciNet  MATH  Google Scholar 

• Barrio RA, Maini PK, Padilla P, Plaza RG, Sánchez-Garduno F (2004) The effect of growth and curvature on pattern formation. J Dyn Diff Equ 4: 1093–1121

  Google Scholar 

• Calhoun DA, Helzel C (2009) A finite volume method for solving parabolic equations on logically cartesian curved surface meshes. SIAM J Sci Comput 6: 4066–4099

  MathSciNet  Google Scholar 

• Chaplain MAJ, Ganesh M, Graham IG (2001) Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J Math Biol 42: 387–423

  Article  MathSciNet  MATH  Google Scholar 

• Crampin EJ, Gaffney EA, Maini PK (2002) Mode doubling and tripling in reation-diffusion patterns on growing domains: a piecewise linear model. J Math Biol 44: 107–128

  Article  MathSciNet  MATH  Google Scholar 

• Crampin EJ, Gaffney EA, Maini PK (1999) Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull Math Biol 61: 1093–1120

  Article  Google Scholar 

• Deckelnick KP, Dziuk G, Elliott CM (2005) Computation of Geometric PDEs and Mean Curvature Flow. Acta Numerica 14: 139–232

  Article  MathSciNet  MATH  Google Scholar 

• Dziuk G (1988) Finite Elements for the Beltrami operator on arbitrary surfaces. Lecture Notes in Mathematics Partial differential equations and calculus of variations, vol 1357. Springer, Berlin, pp 142–155

• Dziuk G, Elliott CM (2007) Finite elements on evolving surfaces. IMA J Num Anal 27: 262–292

  Article  MathSciNet  MATH  Google Scholar 

• Dziuk G, Elliott CM (2007) Surface finite elements for parabolic equations. J Comp Math 25: 430–439

  MathSciNet  Google Scholar 

• Dziuk G, Elliott CM (2010) An Eulerian approach to transport and diffusion on evolving surfaces. Comput Vis Sci 13: 17–28

  Article  MathSciNet  MATH  Google Scholar 

• Dziuk G, Elliott CM (2010) L ² estimates for the evolving surface finite element method. Math Comp (submitted)

• Eilks C, Elliott CM (2008) Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J Comp Phys 227: 9727–9741

  Article  MathSciNet  MATH  Google Scholar 

• Elliott CM, Stinner B (2010) Modeling and computation of two phase geometric biomembranes using surface finite elements. J Comp Phys 229: 6585–6612

  Article  MathSciNet  MATH  Google Scholar 

• Elliott CM, Stinner B, Styles VM (2010) Numerical computation of advection and diffusion on evolving diffuse interfaces. IMA J Num Anal. Advance Access published on 11 May 2010. doi:10.1093/imanum/drq005

• Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12: 30–39

  Article  Google Scholar 

• Golub GH, Van Loan CF (1996) Matrix Computations. JHU Press, Baltimore

  MATH  Google Scholar 

• Greer J, Bertozzi AL, Sapiro G (2006) Fourth order partial differential equations on general geometries. J Comput Phys 216: 216–246

  Article  MathSciNet  MATH  Google Scholar 

• Kondo S, Asai R (1995) A reaction-diffusion wave on the skin of the marine anglefish, Pomacanthus. Nature 376: 765–768

  Article  Google Scholar 

• Lefevre J, Mangin J-F (2010) A reaction-diffusion model of the human brain development. PLoS Comput Biol 6: e1000749

  Article  MathSciNet  Google Scholar 

• Madzvamuse A, Maini PK, Wathen AJ (2003) A moving grid finite element method applied to a model biological pattern generator. J Comp Phys 190: 478–500

  Article  MathSciNet  MATH  Google Scholar 

• Madzvamuse A, Wathen AJ, Maini PK (2005) A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J Sci Comp 24(2): 247–262

  Article  MathSciNet  MATH  Google Scholar 

• Madzvamuse A (2006) Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. J Comput Phys 216: 239–263

  Article  MathSciNet  Google Scholar 

• Madzvamuse A, Maini P (2007) Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains. J Comput Phys 225: 100–119

  Article  MathSciNet  MATH  Google Scholar 

• Madzvamuse A (2009) Turing instability conditions for growing domains with divergence free mesh velocity. Nonlinear Anal Theory Methods Appl 12: 2250–2257

  Google Scholar 

• Madzvamuse A, Gaffney EA, Maini PK (2009) Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. J Math Biol 61: 133–164

  Article  MathSciNet  Google Scholar 

• Maini PK, Painter KJ, Chau HNP (1997) Spatial pattern formation in chemical and biological systems. Faraday Trans 93: 3601–3610

  Article  Google Scholar 

• Murray JD (2002) Mathematical biology I and II, 3rd edn. Springer, Berlin

  Google Scholar 

• Plaza RG, Sánchez-Garduño F, Padilla P, Barrio RA, Maini PK (2004) The effect of growth and curvature on pattern formation. J Dynam Diff Eqs 16(4): 1093–11214

  Article  MATH  Google Scholar 

• Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems. II. J Chem Phys 48: 1695–1700

  Article  Google Scholar 

• Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behaviour. J Theor Biol 81: 389–400

  Article  MathSciNet  Google Scholar 

• Schmidt A, Siebert KG (2005) Design of adaptive finite element software: the finite element toolbox ALBERTA, vol 42. Lecture notes in computational science and engineering. Springer, Berlin

  Google Scholar 

• Turing A (1952) The chemical basis of morphogenesis. Phil Trans R Soc Lond B 237: 37–72

  Article  Google Scholar 

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