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

2019 ◽  
Vol 143 (4) ◽  
pp. 797-853 ◽  
Author(s):  
Balázs Kovács ◽  
Buyang Li ◽  
Christian Lubich
Keyword(s):  
Finite Element ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Closed Surfaces ◽  
Element Algorithm

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.

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 Mean curvature flow with free boundary outside a hypersphere

2017 ◽  
Vol 369 (12) ◽  
pp. 8319-8342 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow
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