The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

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

doi:10.1002/num.20637

Cited by 59 publications

(107 citation statements)

References 37 publications

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“…If the solution

((), {\overset{U}{true}}^{m + 1}, P^{m + 1}, {\overset{X}{true}}^{m + 1}, κ^{m + 1})

to – is such that the interface has not moved, Γ m + 1 =Γ m , then it holds that

\exists ζ \in W ({normalΓ}^{m}) : {(〈〉, ζ {\overset{ν}{true}}^{m}, \overset{η}{true})}_{{normalΓ}^{m}}^{h} + {(〈〉, \nabla_{s} \overset{id}{true}, \nabla_{s} \overset{η}{true})}_{{normalΓ}^{m}} = 0 \forall \overset{η}{true} \in \underset{V}{false} ({normalΓ}^{m}) .

We recall from , Remark 2.4] that in the case d = 2 implies that Γ m is equidistributed, with the possible exception of elements

σ_{j}^{m}

that are locally parallel to each other; see also , Theorem 2.2]. Moreover, we recall from , §4.1] that surfaces

{normalΓ}^{m} \subset {double-struckR}^{3}

that satisfy are called conformal polyhedral surfaces.…”

Section: Methodsmentioning

confidence: 99%

“…We recall from [2, Remark 2.4] that (3.11) in the case d D 2 implies that m is equidistributed, with the possible exception of elements m j that are locally parallel to each other; see also [29,Theorem 2.2]. Moreover, we recall from [3, §4.1] that surfaces m R 3 that satisfy (3.11) are called conformal polyhedral surfaces.…”

Section: Discrete Stationary Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

Fitted finite element discretization of two‐phase Stokes flow

Agnese

Nürnberg

2016

Numerical Methods in Fluids

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We propose a novel fitted finite element method for two-phase Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The method can be shown to be unconditionally stable. Moreover, spherical stationary solutions are captured exactly by the numerical approximation. In addition, the meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm.

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“…If the solution

((), {\overset{U}{true}}^{m + 1}, P^{m + 1}, {\overset{X}{true}}^{m + 1}, κ^{m + 1})

to – is such that the interface has not moved, Γ m + 1 =Γ m , then it holds that

\exists ζ \in W ({normalΓ}^{m}) : {(〈〉, ζ {\overset{ν}{true}}^{m}, \overset{η}{true})}_{{normalΓ}^{m}}^{h} + {(〈〉, \nabla_{s} \overset{id}{true}, \nabla_{s} \overset{η}{true})}_{{normalΓ}^{m}} = 0 \forall \overset{η}{true} \in \underset{V}{false} ({normalΓ}^{m}) .

We recall from , Remark 2.4] that in the case d = 2 implies that Γ m is equidistributed, with the possible exception of elements

σ_{j}^{m}

that are locally parallel to each other; see also , Theorem 2.2]. Moreover, we recall from , §4.1] that surfaces

{normalΓ}^{m} \subset {double-struckR}^{3}

that satisfy are called conformal polyhedral surfaces.…”

Section: Methodsmentioning

confidence: 99%

Section: Discrete Stationary Solutionsmentioning

confidence: 99%

Fitted finite element discretization of two‐phase Stokes flow

Agnese

Nürnberg

2016

Numerical Methods in Fluids

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“…It is to be expected that an arbitrary Lagrange -Euler (ALE) approach combined with the procedure of this paper can substantially mitigate this problem. In the literature this problem has been addressed, for example, by Elliott and Fritz [26] using the DeTurck trick, and by Barrett, Garcke and Nürnberg in a series of papers [4,5,6,7,8]. -The normal vector ν h obtained from the discretized evolution equation is not the same as the normal vector ν Γ h [x] of the discrete surface.…”

Section: Conclusion and Further Commentsmentioning

confidence: 99%

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces

Kovács

Lubich

2019

Numer. Math.

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This paper is dedicated to Gerhard Dziuk on the occasion of his 70th birthday and to Gerhard Huisken on the occasion of his 60th birthday.Abstract A proof of convergence is given for semi-and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk's method, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order H 1 -norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix-vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.Keywords mean curvature flow · geometric evolution equations · evolving surface finite elements · linearly implicit backward difference formula · stability · convergence analysis Mathematics Subject Classification (2000) 35R01 · 65M60 · 65M15 · 65M12

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“…Numerical simulations usually have to be stopped, when the mesh degenerates and some sophisticated machinery for remeshing the polyhedral surfaces, for example, using harmonic maps between surfaces [29], has to be applied. It seems therefore to be a far better solution of this problem to use algorithms that already induce tangential motions that lead to good redistributions of mesh points, see [2,3,4,5,6]. On the other hand, introducing schemes that lead to artificial tangential motions seems to be problematic too.…”

Section: Introduction Motivationmentioning

confidence: 99%

On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick

Elliott¹,

Fritz²

2016

IMA J Numer Anal

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In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.

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The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

Cited by 59 publications

References 37 publications

Fitted finite element discretization of two‐phase Stokes flow

Fitted finite element discretization of two‐phase Stokes flow

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces

On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick

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