4,137 research outputs found
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 , , where 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 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 of a foliation of consisting of uniformly convex hypersurfaces in the direction of their
outer normals with speeds . For quite general functions of the
principal curvatures of the flow hypersurfaces and a smooth and positive
function on (considered as a function of the normal) we show that there
is a distinct leaf in this foliation with the property that
the flow starting from converges to a translating solution of
the flow equation. Furthermore, when starting the flow from a leave inside
it shrinks to a point and when starting the flow from a leave
outside it expands to infinity. While \cite{1} considered this
mechanism with equal to the Gauss curvature we allow to be among others
the elementary symmetric polynomials . 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 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 with normal
speed equal to a power of the mean curvature is considered and the
levelset solution of the flow is obtained as the -limit of a sequence
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 on each
tetraeder of the triangulation. We show in the case (i.e. the evolving
hypersurfaces are curves), that there are solutions of the
above regularized equations in the finite element sense, and estimate the
approximation error between and .
Our method can be extended to the case , if one uses higher order finite
elements.Comment: 20 page
-error estimate for the finite element method on two dimensional surfaces
We approximate the solution of the equation on a
two-dimensional, embedded, orientable, closed surface where
denotes the Laplace Beltrami operator on by using continuous, piecewise
linear finite elements on a triangulation of with flat triangles. We show
that the -error is of order 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 satisfying uniform
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
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