article

Numerical Preservation of Velocity Induced Invariant Regions for Reaction---Diffusion Systems on Evolving Surfaces

Authors:

Massimo Frittelli,

Anotida Madzvamuse,

Chandrasekhar VenkataramanAuthors Info & Claims

Journal of Scientific Computing, Volume 77, Issue 2

Pages 971 - 1000

https://doi.org/10.1007/s10915-018-0741-7

Published: 01 November 2018 Publication History

Abstract

We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction---diffusion systems (RDSs) on surfaces in $${\mathbb {R}}^3$$R3 that evolve under a given velocity field. A fully-discrete method based on the implicit---explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction---diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM---IMEX Euler discretisation of a RDS on a logistically growing surface.

References

[1]

Barreira, R., Elliott, C.M., Madzvamuse, A.: The surface finite element method for pattern formation on evolving biological surfaces. J. Math. Biol. 63(6), 1095---1119 (2011)

[2]

Bertalmıo, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759---780 (2001)

Digital Library

[3]

Blazakis, K.N., Madzvamuse, A., Reyes-Aldasoro, C.C., Styles, V., Venkataraman, C.: Whole cell tracking through the optimal control of geometric evolution laws. J. Comput. Phys. 297, 495---514 (2015)

Digital Library

[4]

Chellaboina, V., Bhat, S.P., Haddad, W.M., Bernstein, D.S.: Modeling and analysis of mass-action kinetics. IEEE Control Syst. 29, 60---78 (2009)

[5]

Chen, C, Kublik, C, Tsai, R.: An implicit boundary integral method for interfaces evolving by Mullins---Sekerka dynamics. In: Maekawa, Y., Jimbo, S. (eds.) Mathematics for Nonlinear Phenomena--Analysis and Computation, pp. 1---21. Springer (2017)

[6]

Chueh, K.N., Conley, C.C., Smoller, J.A.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26, 373---392 (1977)

[7]

Corson, F., Hamant, O., Bohn, S., Traas, J., Boudaoud, A., Couder, Y.: Turning a plant tissue into a living cell froth through isotropic growth. Proc. Nat. Acad. Sci. 106(21), 8453---8458 (2009)

[8]

Crampin, E.J., Gaffney, E.A., Maini, P.K.: Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61(6), 1093---1120 (1999)

[9]

Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262---292 (2007)

[10]

Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289---396 (2013)

[11]

Dziuk, G., Elliott, C.M.: L$$^2$$2-estimates for the evolving surface finite element method. Math. Comput. 82(281), 1---24 (2013)

[12]

Elliott, C.M., Stinner, B., Venkataraman, C.: Modelling cell motility and chemotaxis with evolving surface finite elements. J. R. Soc. Interface 9(76), 3027---3044 (2012)

[13]

Elliott, C.M., Venkataraman, C.: Error analysis for an ALE evolving surface finite element method. Numer. Methods Partial Differ. Equ. 31(2), 459---499 (2015)

[14]

Frittelli, M: Numerical Methods for Partial Differential Equations on Stationary and Evolving Surfaces. Ph.D. thesis, Dipartimento di Matematica e Fisica "E. De Giorgi", Università del Salento (2018) (to appear)

[15]

Frittelli, M., Madzvamuse, A., Sgura, I., Venkataraman, C.: Lumped finite elements for reaction-cross-diffusion systems on stationary surfaces. Comput. Math. Appl. 74(12), 3008---3023 (2017).

Digital Library

[16]

Frittelli, M., Madzvamuse, A., Sgura, I., Venkataraman, C.: Preserving invariance properties of reaction---diffusion systems on stationary surfaces. IMA J. Numer. Anal. (2017).

[17]

Frittelli, M., Sgura, I.: Virtual element method for the Laplace---Beltrami equation on surfaces. ESAIM Math. Model. Numer. Anal. (2017).

[18]

Fuselier, E.J., Wright, G.B.: A high-order kernel method for diffusion and reaction---diffusion equations on surfaces. J. Sci. Comput. 56(3), 535---565 (2013)

[19]

González-Olivares, E., Ramos-Jiliberto, R.: Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol. Model. 166, 135---146 (2003)

[20]

Grande, J., Reusken, A.: A higher order finite element method for partial differential equations on surfaces. SIAM J. Numer. Anal. 54(1), 388---414 (2016)

[21]

Hebey, E., Robert, F.: Sobolev spaces on manifolds. In: Krupka, D., Saunders, D. (eds.) Handbook of Global Analysis, pp. 375---415 (2008)

[22]

Kovács, B.: Computing arbitrary Lagrangian Eulerian maps for evolving surfaces. arXiv preprint arXiv:1612.01701 (2016)

[23]

Kovács, B., Li, B., Lubich, C., Guerra, C.A.P.: Convergence of finite elements on an evolving surface driven by diffusion on the surface. Numer. Math. 137(3), 643---689 (2017)

Digital Library

[24]

Lacitignola, D., Bozzini, B., Frittelli, M., Sgura, I.: Turing pattern formation on the sphere for a morphochemical reaction---diffusion model for electrodeposition. Commun. Nonlinear Sci. Numer. Simul. 48, 484---508 (2017).

[25]

Lacitignola, D., Bozzini, B., Sgura, I.: Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay. Eur. J. Appl. Math. 26(2), 143---173 (2015)

[26]

Lefèvre, J., Mangin, J.F.: A reaction---diffusion model of human brain development. PLoS Comput. Biol. 6(4), e1000,749 (2010)

[27]

Li, H., Lin, Y., Heath, R.M., Zhu, M.X., Yang, Z.: Control of pollen tube tip growth by a rop gtpase-dependent pathway that leads to tip-localized calcium influx. Plant Cell 11(9), 1731---1742 (1999)

[28]

Madzvamuse, A., Barreira, R.: Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces. Phys. Rev. E 90(4), 043,307 (2014)

[29]

Madzvamuse, A., Gaffney, E.A., Maini, P.K.: Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. J. Math. Biol. 61(1), 133---164 (2010)

[30]

Madzvamuse, A., Wathen, A.J., Maini, P.K.: A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys. 190(2), 478---500 (2003)

Digital Library

[31]

Murray, J.D.: Mathematical Biology. II Spatial Models and Biomedical Applications (Interdisciplinary Applied Mathematics), vol. 18. Springer, New York (2001)

[32]

Plaza, R.G., Sanchez-Garduno, F., Padilla, P., Barrio, R.A., Maini, P.K.: The effect of growth and curvature on pattern formation. J. Dyn. Differ. Equ. 16(4), 1093---1121 (2004)

[33]

Pruyne, D., Bretscher, A.: Polarization of cell growth in yeast. I. Establishment and maintenance of polarity states. J. Cell Sci. 113(3), 365---375 (2000)

[34]

Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, vol. 23. Springer, New York (2008)

Digital Library

[35]

Ren, K., Tsai, R., Zhong, Y.: An implicit boundary integral method for computing electric potential of macromolecules in solvent. arXiv:1709.08070v4 (2018)

[36]

Sahlin, P., Jönsson, H.: A modeling study on how cell division affects properties of epithelial tissues under isotropic growth. PLoS ONE 5(7), e11,750 (2010)

[37]

Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81(3), 389---400 (1979)

[38]

Smoller, J.: Shock Waves and Reaction Diffusion Equations. Springer, New York (1994)

[39]

Taylor, M.E.: Partial Differential Equations. III. Springer, New York (2011)

[40]

Tuncer, N., Madzvamuse, A.: Projected finite elements for systems of reaction---diffusion equations on closed evolving spheroidal surfaces. Commun. Comput. Phys. 21(3), 718---747 (2017)

[41]

Venkataraman, C., Lakkis, O., Madzvamuse, A.: Global existence for semilinear reaction---diffusion systems on evolving domains. J. Math. Biol. 64(1---2), 41---67 (2012)

[42]

Yang, F.W., Venkataraman, C., Styles, V., Madzvamuse, A.: A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws. Commun. Comput. Phys. 21(1), 65---92 (2017)

Cited By

Frittelli MSgura I(2024)Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domainsApplied Numerical Mathematics10.1016/j.apnum.2023.07.010200:C(286-308)Online publication date: 1-Jun-2024
https://dl.acm.org/doi/10.1016/j.apnum.2023.07.010
D’Autilia MSgura ISimoncini V(2020)Matrix-oriented discretization methods for reaction–diffusion PDEsComputers & Mathematics with Applications10.1016/j.camwa.2019.10.02079:7(2067-2085)Online publication date: 1-Apr-2020
https://dl.acm.org/doi/10.1016/j.camwa.2019.10.020

Numerical Preservation of Velocity Induced Invariant Regions for Reaction---Diffusion Systems on Evolving Surfaces
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

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Published In

cover image Journal of Scientific Computing

Journal of Scientific Computing Volume 77, Issue 2

November 2018

614 pages

ISSN:0885-7474

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Copyright © Copyright © 2018 Springer Science+Business Media, LLC, part of Springer Nature.

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Plenum Press

United States

Publication History

Published: 01 November 2018

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Cited By

Frittelli MSgura I(2024)Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domainsApplied Numerical Mathematics10.1016/j.apnum.2023.07.010200:C(286-308)Online publication date: 1-Jun-2024
https://dl.acm.org/doi/10.1016/j.apnum.2023.07.010
D’Autilia MSgura ISimoncini V(2020)Matrix-oriented discretization methods for reaction–diffusion PDEsComputers & Mathematics with Applications10.1016/j.camwa.2019.10.02079:7(2067-2085)Online publication date: 1-Apr-2020
https://dl.acm.org/doi/10.1016/j.camwa.2019.10.020

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