scholarly journals Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs

2000 ◽  
pp. 341-359 ◽  
Author(s):  
Klaus Deckelnick ◽  
Gerhard Dziuk
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
The Mean ◽  
Discrete Finite Element

scholarly journals Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mouhamadou Samsidy Goudiaby ◽  
Ababacar Diagne ◽  
Leon Matar Tine
Keyword(s):  
Finite Element ◽  
Liquid Crystal ◽  
Type Inequality ◽  
Mesh Size ◽  
Time Step ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
Discrete Finite Element ◽  
Longtime Behavior

<p style='text-indent:20px;'>We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.</p>

scholarly journals A finite element error analysis for axisymmetric mean curvature flow

2020 ◽  
Author(s):  
John W Barrett ◽  
Klaus Deckelnick ◽  
Robert Nürnberg
Keyword(s):  
Finite Element ◽  
Mean Curvature ◽  
Numerical Approximation ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Optimal Error ◽  
Fully Discrete ◽  
Fully Discrete Approximation ◽  
Element Error ◽  
Theoretical Results

Abstract We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface we derive optimal error bounds with respect to the $L^2$- and $H^1$-norms for a fully discrete approximation. We perform convergence experiments to confirm the theoretical results and also present numerical simulations for some genus-0 and genus-1 surfaces, including for the Angenent torus.

Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy

2015 ◽  
Vol 53 (1-2) ◽  
pp. 413-443 ◽  
Author(s):  
Vimal Srivastava ◽  
Sudhakar Chaudhary ◽  
V. V. K. Srinivas Kumar ◽  
Balaji Srinivasan
Keyword(s):  
Finite Element ◽  
Parabolic Problem ◽  
Dirichlet Energy ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
Discrete Finite Element

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

scholarly journals Finite topology self-translating surfaces for the mean curvature flow in R3

2017 ◽  
Vol 320 ◽  
pp. 674-729 ◽  
Author(s):  
Juan Dávila ◽  
Manuel del Pino ◽  
Xuan Hien Nguyen
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Finite Topology ◽  
The Mean

The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ 2 {\mathbb{C}^{2}}

2018 ◽  
Vol 2018 (743) ◽  
pp. 229-244 ◽  
Author(s):  
Jingyi Chen ◽  
John Man Shun Ma
Keyword(s):  
Upper Bound ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Riemannian Metric ◽  
Branch Points ◽  
Rigidity Result ◽  
The Mean ◽  
Uniform Area

Abstract Let F_{n} : (Σ, h_{n} ) \to \mathbb{C}^{2} be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics \{ h_{n} \} converges smoothly to a Riemannian metric h. We show that a subsequence of \{ F_{n} \} converges smoothly to a branched conformally immersed Lagrangian self-shrinker F_{\infty} : (Σ, h) \to \mathbb{C}^{2} . When the area bound is less than 16π, the limit {F_{\infty}} is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence h_{n} \to h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.

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