Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case

Pozzi, Paola; Stinner, Björn

doi:10.1093/imanum/dry004

Cited by 6 publications

(6 citation statements)

References 38 publications

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“…Such an approach using the angle function has been used for instance in [20] to prove well-posedness and global existence for a flow towards elastica. Local and global existence of solutions for the classical elastic flow, given by a parabolic fourth order equation in R n , has been shown in several works, for instance [10,16,8,7,15,21,6,19,17,18]. For a more detailed overview, see the recent survey [14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A dynamic approach to heterogeneous elastic wires

Dall’Acqua¹,

Langer²,

Rupp³

2022

Preprint

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We consider closed planar curves with fixed length whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated L 2 -gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness and global existence of solutions for initial data in a weak regularity class and with arbitrary winding number.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A dynamic approach to heterogeneous elastic wires

Dall’Acqua¹,

Langer²,

Rupp³

2022

Preprint

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“…The case of open curves with a fix boundary was analysed in [SVY20]. For forced-elastic flow of curves semi-discrete error estimates were proved in [PS19]. For mean curvature flow coupled to a diffusion process on a graph optimal-order fully discrete error bounds were recently shown in [DS21].…”

Section: Related Numerical Analysismentioning

confidence: 99%

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

Elliott¹,

Garcke²,

Kovács³

2022

Preprint

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An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction-diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the H 1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.

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“…The case of open curves with a fix boundary was analysed in [49]. For forced-elastic flow of curves semi-discrete error estimates were proved in [48]. For mean curvature flow coupled to a diffusion process on a graph optimal-order fully discrete error bounds were recently shown in [24].…”

Section: Related Numerical Analysismentioning

confidence: 99%

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

2022

View full text Add to dashboard Cite

An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $$H^1$$ H 1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.

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Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case

Cited by 6 publications

References 38 publications

A dynamic approach to heterogeneous elastic wires

A dynamic approach to heterogeneous elastic wires

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

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