Parametric finite element approximations of curvature-driven interface evolutions

Barrett, John W.; Garcke, Harald; Nürnberg, Robert

doi:10.1016/bs.hna.2019.05.002

Cited by 51 publications

(60 citation statements)

References 150 publications

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“…are evolving hypersurfaces (see, e. g., [6,Def. 23]) that do not intersect with ∂Ω and we define Σ( , t) := {x ∈ Ω : ϕ (x, t) = 0}.…”

Section: Constitutive Assumptionsmentioning

confidence: 99%

“…In the following we denote by C a generic constant independent of δ ∈ (0, δ 0 ]. Using (5.24a) and elliptic regularity theory, it follows thatϕ δ L ∞ (H 1 )∩L 2 (H 2 ) ≤ C.(5.32)Due to (5.24a) we havev δ L 2 (H 1 ) + √ δ v δ L ∞ (L 2 ) ≤ C. (5.33)Next, multiplying (5.8d) 1 with σ 5 δ , integrating by parts and using that∇σ δ • v δ , σ 5 δ = 1 6 ∇(|σ δ | 6 ) , v δ = − 1 6 |σ δ |6 , div(v δ ) = 0 f. a. e. t ∈ (0,…”

mentioning

confidence: 99%

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Cahn–Hilliard–Brinkman systems for tumour growth

Ebenbeck¹,

Garcke²,

Nürnberg³

2021

DCDS-S

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A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn-Hilliard equation for the evolution of the tumour to a reactiondiffusion-advection equation for a nutrient and to a Brinkman-Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.

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“…are evolving hypersurfaces (see, e. g., [6,Def. 23]) that do not intersect with ∂Ω and we define Σ( , t) := {x ∈ Ω : ϕ (x, t) = 0}.…”

Section: Constitutive Assumptionsmentioning

confidence: 99%

mentioning

confidence: 99%

Cahn–Hilliard–Brinkman systems for tumour growth

Ebenbeck¹,

Garcke²,

Nürnberg³

2021

DCDS-S

Self Cite

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show abstract

“…For completeness, let us mention that there are also interface capturing approaches that avoid the need P. Pozzi & B. Stinner to look after the mesh quality [7,23,28,5,17]. Such approaches comprise phase field models and level set methods, for overviews we refer to [11,4,13,27].…”

Section: P Pozzi and B Stinnermentioning

confidence: 99%

On motion by curvature of a network with a triple junction

Pozzi¹,

Stinner²

2021

The SMAI Journal of Computational Mathematics

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We numerically study the planar evolution by curvature flow of three parametrised curves that are connected by a triple junction in which conditions are imposed on the angles at which the curves meet. One of the key problems in analysing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularisation parameter shrinks to zero. Convergence of the new semi-discrete finite element scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularisation parameter on the properties of scheme and the accuracy of the results is numerically investigated.

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“…Differential geometry The notation of this paragraph follows [4] and [6]. With Diff q (X, Y ) we denote the set of all C q -diffeomorphisms mapping the Banach space X into the Banach space Y .…”

Section: Preliminariesmentioning

confidence: 99%

A PDE model for bleb formation and interaction with linker proteins

Werner

Burger

Pietschmann

2020

Transactions of Mathematics and Its Applications

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The aim of this paper is to further develop mathematical models for bleb formation in cells, including cell-membrane interactions with linker proteins. This leads to nonlinear reaction-diffusion equations on a surface coupled to fluid dynamics in the bulk. We provide a detailed mathematical analysis and investigate some singular limits of the model, connecting it to previous literature. Moreover, we provide numerical simulations in different scenarios, confirming that the model can reproduce experimental results on bleb initation. Cell blebbing, Surface PDEs, Fluid-Structure Interaction, Free Boundary Problems. T for the symmetrised gradient. The set of k-times, k ≥ 0, weak differentiable, X-valued functions, where X is a vector space, on some open set Ω ⊆ R 3 is H k (Ω, X). We will also employ the subspaces of mean-value-free functions:of functions with time-constant mean value:T ∈ [0, ∞), of solenoidals functions:, and the traces of solenoidal functions:

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Parametric finite element approximations of curvature-driven interface evolutions

Cited by 51 publications

References 150 publications

Cahn–Hilliard–Brinkman systems for tumour growth

Cahn–Hilliard–Brinkman systems for tumour growth

On motion by curvature of a network with a triple junction

A PDE model for bleb formation and interaction with linker proteins

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