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

Frittelli, Massimo; Madzvamuse, Anotida; Sgura, Ivonne; Venkataraman, Chandrasekhar

doi:10.1007/s10915-018-0741-7

Cited by 12 publications

(12 citation statements)

References 39 publications

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“…Among them, an algorithm of ALE maps with angle conditions is also described in detail and illustrated as well. Such methods are of interest for qualitative results, such as DMPs [24][25][26]33].…”

Section: Resultsmentioning

confidence: 99%

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Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

Kovács

2018

Numerical Methods Partial

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The good mesh quality of a discretized closed evolving surface is often compromised during time evolution. In recent years this phenomenon has been theoretically addressed in a few ways, one of them uses arbitrary Lagrangian Eulerian (ALE) maps. However, the numerical computation of such maps still remained an unsolved problem in the literature. An approach, using differential algebraic problems, is proposed here to numerically compute an arbitrary Lagrangian Eulerian map, which preserves the mesh properties over time. The ALE velocity is obtained by finding an equilibrium of a simple spring system, based on the connectivity of the nodes in the mesh. We also consider the algorithmic question of constructing acute surface meshes. We present various numerical experiments illustrating the good properties of the obtained meshes and the low computational cost of the proposed approach.

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Section: Resultsmentioning

confidence: 99%

“…In a couple of recent works discrete maximum principles (DMPs) and invariant regions have been studied for surface PDEs discretized by surface finite elements, see . It is well known that acute or nonobtuse meshes are required for DMPs even for flat domains, and also for triangulated surface meshes.…”

Section: Possible Extensionsmentioning

confidence: 99%

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

Kovács

2018

Numerical Methods Partial

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“…Numerical investigations. The formulation (6) lends itself to to space discretization through the Lumped Evolving Surface Finite Element Method (LES-FEM), see [12], which we will adopt in the present work. Among the numerous space discretisation techniques for surface RDSs existing in the literature, LESFEM does not rely on a parametrization of the surface.…”

Section: Effects Of Growth On the Turing Regionsmentioning

confidence: 99%

“…Let us now consider reaction-diffusion systems on evolving surfaces, by following the presentation in [12]. Let Γ 0 be a C 2 surface without boundary in R 3 .…”

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Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

Lacitignola

Frittelli

Cusimano

et al. 2022

JCD

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<p style='text-indent:20px;'>In this study, the formation of the adult sea urchin shape is rationalized within the Turing's theory paradigm. The emergence of protrusions from the expanding underlying surface is described through a reaction-diffusion model with Gray-Scott kinetics on a growing oblate spheroid. The case of slow exponential isotropic growth is considered. The model is first studied in terms of the spatially homogenous equilibria and of the bifurcations involved. Turing diffusion-driven instability is shown to occur and the impact of the slow exponential growth on the resulting Turing regions adequately discussed. Numerical investigations validate the theoretical results showing that the combination between an inhibitor and an activator can result in a distribution of spot concentrations that underlies the development of ambulacral tentacles in the sea urchin's adult stage. Our findings pave the way for a model-driven experimentation that could improve the current biological understanding of the gene control networks involved in patterning.</p>

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Virtual element method for elliptic bulk‐surface PDEs in three space dimensions

Frittelli

Madzvamuse

Sgura

2023

Numerical Methods Partial

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In this work we present a novel bulk‐surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk‐surface partial differential equations in three space dimensions. The BSVEM is based on the discretization of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. We present a geometric error analysis of bulk‐surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second‐order convergence in space, provided the exact solution is in the bulk and on the surface, where the additional is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. Two numerical examples on the unit sphere and on the Dupin ring cyclide confirm the convergence result.

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

Cited by 12 publications

References 39 publications

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

Virtual element method for elliptic bulk‐surface PDEs in three space dimensions

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