Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Abstract

Article (Published Version) http://sro.sussex.ac.uk Madzvamuse, A and Barreira, R (2014) Exhibiting cross-diffusion-induced patterns for reactiondiffusion systems on evolving domains and surfaces. Physical Review E, 90. 043307-1-043307-14. ISSN 1539-3755 This version is available from Sussex Research Online: http://sro.sussex.ac.uk/51334/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and reuse:Sussex Research Online is a digital repository of the research output of the University.Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.PHYSICAL REVIEW E 90, 043307 (2014) Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in...

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…For k << 1 (in particular when k gets closer to zero), the Coupled bulk-surface reaction-diffusion systems parameter spaces become almost identical and these represent parameter spaces in the absence of domain growth. As k becomes larger and larger, the parameter spaces become more distinct and larger in size (including the circular subregions), reinforcing earlier results for standard reaction-diffusion systems on growing domains [21,23,25,27]. The disconnected parameter spaces emerge due to the increase in the growth rate k as well as due to the dimension m.…”

“…It must be observed that for the fractional-step θ method above, the computational mesh is continuously evolving according to the exponential growth and all matrices are assembled at different discretised surfaces depending on the time-level n. The number of degrees of freedom and the mesh connectivity remains constant throughout domain growth. For further details on the implementation of the evolving (bulk) surface finite element method we refer the interested reader to [2,6,7,23].…”

Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces

“…where the constant above is independent of t. ii. Let L(u) be (26), let f ∈ L 2 ( (t)) and let u ∈ H 1 ( (t)) be a weak solution of (27). Then u is a strong solution of (27), that is, u solves (27) almost everywhere and there exists a constant c > 0 independent of t and u such that ||u|| H 2 ( (t)) ≤ c(||u|| L 2 ( (t)) + ||f || L 2 ( (t)) ).…”

“…A great number of real-life phenomena are modeled by nonlinear parabolic problems on evolving surfaces. Apart from general quasilinear problems on moving surfaces, see for example, 3.5 in [15], more specific applications are the nonlinear models: diffusion induced grain boundary motion [16][17][18][19][20]; Allen-Cahn and Cahn-Hilliard equations on evolving surfaces [1,[21][22][23][24]; modeling solid tumor growth [20,25]; pattern formation modeled by reaction-diffusion equations [26,27]; image processing [28]; Ginzburg-Landau model for superconductivity [29].…”

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

“…In many applications the spatial domain is a curved surface rather than a flat domain, which may be time-dependent. Among the applications of RDSs on surfaces we mention brain growth [26], cell migration [3], chemotaxis [12], developmental biology [28], electrodeposition [24] and phase field modeling [42]. The growing interest toward PDEs on evolving surfaces has stimulated the development of several numerical methods for such problems, among which we mention (but not limited to) embedding methods [2], kernel methods [18], implicit boundary integral methods [5,35], surface finite element methods (SFEM) [10] and some of their recent variations and extensions [13,16,17,20,23,40].…”