Lumped finite elements for reaction-cross-diffusion systems on stationary surfaces

All the authors (AM, IS, CV, MF) thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme (Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation; EPSRC EP/K032208/1). This work (AM) has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642866. AM and CV acknowledge support from the Engineering and Physical Sciences Research Council (EP/J016780/1) on Modelling, analysis and simulation of spatial patterning on evolving biological surfaces and the Leverhulme Trust Research Project Grant (RPG-2014-149) on Unravelling new mathematics for 3D cell migration. AM was partially supported by a fellowship from the Simons Foundation. AM is a Royal Society Wolfson Research Merit Award Holder, generously funded by the Wolfson Foundation.

[1]  M. Schweizer,et al.  Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes , 2005, math/0505208.

[2]  Vidar Thomée,et al.  A Lumped Mass Finite-element Method with Quadrature for a Non-linear Parabolic Problem , 1985 .

[3]  Mark A. J. Chaplain,et al.  Robust numerical methods for taxis-diffusion-reaction systems: Applications to biomedical problems , 2006, Math. Comput. Model..

[4]  C. Venkataraman,et al.  Lumped finite element method for reaction-diffusion systems on compact surfaces , 2016, 1609.02741.

[5]  Irving R Epstein,et al.  Cross-diffusion and pattern formation in reaction-diffusion systems. , 2009, Physical chemistry chemical physics : PCCP.

[6]  Weizhang Huang,et al.  Maximum principle for the finite element solution of time‐dependent anisotropic diffusion problems , 2013 .

[7]  Charles M. Elliott,et al.  Finite element methods for surface PDEs* , 2013, Acta Numerica.

[8]  Zoltán Horváth,et al.  On Preservation of Positivity in Some Finite Element Methods for the Heat Equation , 2015, Comput. Methods Appl. Math..

[9]  J. Smoller Shock Waves and Reaction-Diffusion Equations , 1983 .

[10]  Qiang Du,et al.  Finite element approximation of the Cahn–Hilliard equation on surfaces , 2011 .

[11]  Swarup Poria,et al.  Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity , 2016 .

[12]  Anotida Madzvamuse,et al.  Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations , 2015, Journal of mathematical biology.

[13]  G. Dziuk Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .

[14]  David Hoff,et al.  Stability and Convergence of Finite Difference Methods for Systems of Nonlinear Reaction-Diffusion Equations , 1978 .

[15]  Sergey Korotov,et al.  Discrete maximum principles for nonlinear parabolic PDE systems , 2012 .

[16]  Rangarajan Sudarsan,et al.  A Mixed-Culture Biofilm Model with Cross-Diffusion , 2015, Bulletin of Mathematical Biology.

[17]  Jianxian Qiu,et al.  Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems , 2012, Numerische Mathematik.

[18]  O. Annunziata,et al.  Cross-diffusion in a colloid-polymer aqueous system , 2013 .

[19]  A. Vergara,et al.  Lysozyme Mutual Diffusion in Solutions Crowded by Poly(ethylene glycol) , 2006 .

[20]  Alan J. Laub,et al.  Matrix analysis - for scientists and engineers , 2004 .

[21]  Ricardo H. Nochetto,et al.  Combined effect of explicit time-stepping and quadrature for curvature driven flows , 1996 .

[22]  Christian Lubich,et al.  Variational discretization of wave equations on evolving surfaces , 2014, Math. Comput..

[23]  Emmanuel Hebey,et al.  Sobolev Spaces on Riemannian Manifolds , 1996 .

[24]  Marcus R. Garvie,et al.  Numerische Mathematik Finite element approximation of spatially extended predator – prey interactions with the Holling type II functional response , 2007 .

[25]  Ansgar Jüngel,et al.  Cross Diffusion Preventing Blow-Up in the Two-Dimensional Keller-Segel Model , 2011, SIAM J. Math. Anal..

[26]  Charles M. Elliott,et al.  L2-estimates for the evolving surface finite element method , 2012, Math. Comput..

[27]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[28]  Rodrigo Ramos-Jiliberto,et al.  Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability , 2003 .

[29]  Bath Ba THE GLOBAL DYNAMICS OF DISCRETE SEMILINEAR PARABOLIC EQUATIONS , 1993 .

[30]  Charles M. Elliott,et al.  Evolving surface finite element method for the Cahn–Hilliard equation , 2013, Numerische Mathematik.

[31]  Omar Lakkis,et al.  Implicit-Explicit Timestepping with Finite Element Approximation of Reaction-Diffusion Systems on Evolving Domains , 2011, SIAM J. Numer. Anal..

[32]  Michael E. Taylor,et al.  Partial Differential Equations III , 1996 .

[33]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.