A multigrid finite element method for reaction-diffusion systems on surfaces

Landsberg, Christoph; Voigt, Axel

doi:10.1007/s00791-010-0136-2

Cited by 27 publications

(19 citation statements)

References 38 publications

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“…Numerical method for RD patterns on the sphere. Previous numerical work for RD systems on surfaces include [42], [8], [17], [13], and [3]. For the sphere, in [42] a discretization of the Laplacian in spherical coordinates (θ, φ) was used together with an explicit Euler integration in time.…”

Section: Platonic Solidmentioning

confidence: 99%

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The Stability of Localized Spot Patterns for the Brusselator on the Sphere

Rozada¹,

Ruuth²,

Ward³

2014

SIAM J. Appl. Dyn. Syst.

142

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In the singularly perturbed limit of an asymptotically small diffusivity ratio ε 2 , the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reactiondiffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns and to formulate eigenvalue problems governing the stability of spot patterns to three types of "fast" O(1) time-scale instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of ν = −1/ log ε. From a leading-order-in-ν analysis, and with an asymptotically large inhibitor diffusivity, some rigorous results for competition and oscillatory instabilities are obtained from an analysis of a new class of nonlocal eigenvalue problem (NLEP). Theoretical results for the stability of spot patterns are confirmed with full numerical computations of the Brusselator PDE system on the sphere using the closest point method. [39] proposed that localized peaks in the concentration of a chemical substance, known as a morphogen, could be responsible for the process of morphogenesis, which describes the development of a complex organism from a single cell. By means of a linearized analysis, he showed how stable spatially inhomogeneous patterns can develop from small perturbations of spatially homogeneous initial data for a coupled system of reaction-diffusion (RD) equations. Introduction. Turing inMotivated by this initial work, a systematic and rigorous approach has been developed over the last few decades for the analysis of small amplitude patterns in RD systems. After linearizing the RD system around a spatially uniform state to identify bifurcation points for the emergence of spatially nonuniform patterns, a weakly nonlinear analysis based on a

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Section: Platonic Solidmentioning

confidence: 99%

“…[8], [17], [3], [24]) motivated by specific scientific applications, including those in [28], [15], and [30]. However, to date there has only been limited analytical work on analyzing RD patterns on surfaces.…”

mentioning

confidence: 99%

The Stability of Localized Spot Patterns for the Brusselator on the Sphere

Rozada¹,

Ruuth²,

Ward³

2014

SIAM J. Appl. Dyn. Syst.

142

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“…The reasoning behind this statement is that our methodology based on triangulated surfaces formulates the problem in one dimension less than other approaches, in which the equations are discretized in the embedding space. See, for example [13][14][15][16] and references therein for further discussion.…”

Section: Introductionmentioning

confidence: 99%

Modelling cell motility and chemotaxis with evolving surface finite elements

Elliott

Stinner

Venkataraman

2012

J. R. Soc. Interface.

123

125

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We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction -diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/ maskae/CV_Warwick/Chemotaxis.html.

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“…PDEs need to be solved intrinsically and numerically for data defined on 3D surfaces in many applications. For instance, such examples exist in texture synthesis [35,36], weathering [14,25], computer graphics [5] and surface processing [12,20,21,24,30,31,40]. Usually, surfaces are presented by triangular or polygonal forms.…”

Section: Introductionmentioning

confidence: 99%

A new intrinsic numerical method for PDE on surfaces

Wu¹,

Chi²,

Chen³

2012

International Journal of Computer Mathematics

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In this note, we introduce a simple, effective numerical method, the local tangential lifting method, for solving partial differential equations for scalar-and vector-valued data defined on surfaces. Even though we follow the traditional way to approximate the regular surfaces under consideration by triangular meshes, the key idea of our algorithm is to develop an intrinsic and unified way to compute directly the partial derivatives of functions defined on triangular meshes. We present examples in computer graphics and image processing applications.

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A multigrid finite element method for reaction-diffusion systems on surfaces

Cited by 27 publications

References 38 publications

The Stability of Localized Spot Patterns for the Brusselator on the Sphere

The Stability of Localized Spot Patterns for the Brusselator on the Sphere

Modelling cell motility and chemotaxis with evolving surface finite elements

A new intrinsic numerical method for PDE on surfaces

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