Numerical approximation of level set power mean curvature flow

In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level set formulation of this flow and discretize the regularized level set equation with finite elements. In a previous paper we proved an a priori estimate for the approximation error between the finite element solution and the solution of the original level set equation. We obtained an upper bound for this error which is polynomial in the discretization parameter and the reciprocal regularization parameter. The aim of the present paper is the numerical study of the behavior of the evolution and the numerical verification of certain convergence rates. We restrict the consideration to the case that the level set function depends on two variables, i.e. the moving hypersurfaces are curves. Furthermore, we confirm for specific initial curves and different values of $k$ that the flow improves the isoperimetrical deficit

A note on expansion of convex plane curves via inverse curvature flow

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of the arc length between its endpoints and elapsed time. They apply this estimate to short segments and deduce directly that the maximum curvature decays exponentially to the curvature of a circle with the same length. We consider the expansion of convex curves under inverse (mean) curvature flow and show that the above comparison function also works in this case to obtain a new proof of the fact that the flow exists for all times and becomes round in shape, i.e. converges smoothly to the unit circle after an appropriate rescaling.Comment: 9 page

Error estimate for a finite element approximation of the solution of a linear parabolic equation on a two-dimensional surface

We show that a certain error estimate for a fully discrete finite element approximation of the solution of the heat equation which is defined in a two-dimensional Euclidean domain carries over to the case of a general linear parabolic equation which is defined on a two-dimensional surface

Flowing the leaves of a foliation with normal speed given by the logarithm of general curvature functions

Generalizing results of Chou and Wang \cite{1} we study the flows of the leaves $(M_{\Theta})_{\Theta>0}$ of a foliation of $\mathbb{R}^{n+1}\setminus \{0\}$ consisting of uniformly convex hypersurfaces in the direction of their outer normals with speeds $-\log(F/f)$. For quite general functions $F$ of the principal curvatures of the flow hypersurfaces and $f$ a smooth and positive function on $S^n$ (considered as a function of the normal) we show that there is a distinct leaf $M_{\Theta_{*}}$ in this foliation with the property that the flow starting from $M_{\Theta_{*}}$ converges to a translating solution of the flow equation. Furthermore, when starting the flow from a leave inside $M_{\Theta_{*}}$ it shrinks to a point and when starting the flow from a leave outside $M_{\Theta_{*}}$ it expands to infinity. While \cite{1} considered this mechanism with $F$ equal to the Gauss curvature we allow $F$ to be among others the elementary symmetric polynomials $H_k$. We, furthermore, show that such kind of behavior is robust with respect to relaxing certain assumptions at least in the rotationally symmetric and homogeneous degree one curvature function case.Comment: 34 page

Numerical approximation of positive power curvature flow via deterministic games

We approximate the level set solution for the motion of an embedded closed curve in the plane with normal speed \max(0, \kappa)^{\ga} where $\kappa$ is the curvature of the curve and \frac{1}{3}<\ga<1 by the value functions of a family of deterministic two person games. We show convergence of the value functions to the viscosity solution of the level set equation and propose a numerical scheme for the calculation of the value function.Comment: Presentation of the algorithm corrected, numerical examples adde

A note on inverse mean curvatrue flow in cosmological spacetimes

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in [8] that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface. We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition

Finite element approximation of power mean curvature flow

In [21] the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal speed equal to a power $k>1$ of the mean curvature is considered and the levelset solution $u$ of the flow is obtained as the $C^0$-limit of a sequence $u^{\epsilon}$ of smooth functions solving the regularized levelset equations. We prove a rate for this convergence. Then we triangulate the domain by using a tetraeder mesh and consider continuous finite elements, which are polynomials of degree $\le 2$ on each tetraeder of the triangulation. We show in the case $n=1$ (i.e. the evolving hypersurfaces are curves), that there are solutions $u^{\epsilon}_h$ of the above regularized equations in the finite element sense, and estimate the approximation error between $u^{\epsilon}_h$ and $u$. Our method can be extended to the case $n>1$, if one uses higher order finite elements.Comment: 20 page

L∞L^{\infty}-error estimate for the finite element method on two dimensional surfaces

We approximate the solution of the equation $$-\Delta_S u+u = f$$ on a two-dimensional, embedded, orientable, closed surface $S$ where $-\Delta_S$ denotes the Laplace Beltrami operator on $S$ by using continuous, piecewise linear finite elements on a triangulation of $S$ with flat triangles. We show that the $L^{\infty}$-error is of order $O(h^2|\log h|)$ as in the corresponding situation in an Euclidean setting.Comment: Remark 1.1 adde

Alternative to evolving surface finite element method

ESFEM is a method introduced in order to solve a linear advection-diffusion equation on an evolving two-dimensional surface with finite elements by using a moving grid with nodes sitting on and evolving with the surface. The evolution of the surface is assumed to be given as a smooth one-parameter family of embeddings of a fixed initial surface into $\mathbb{R}^3$ satisfying uniform $C^4$ bounds. We calculate an equivalent transformed equation which is defined on the fixed initial surface and can hence be solved numerically on a fixed grid. We present numerical examples which indicate that both approaches are essentially of the same accuracy

Variational discretization of parabolic control problems on evolving surfaces with pointwise state constraints

We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time. We prove error bounds for control and state.Comment: 13 page