Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature
File(s)
Accepted version
Author(s)
Barrett, JW
Garcke, H
Nurnberg, R
Type
Journal Article
Abstract
We study numerical approximations for geometric evolution
equations arising as
gradient flows for energy functionals that are quadratic in t
he principal curvatures
of a two-dimensional surface. Beside the well-known Willmo
re and Helfrich flows we
will also consider flows involving the Gaussian curvature of
the surface. Boundary
conditions for these flows are highly nonlinear, and we use a v
ariational approach to
derive weak formulations, which naturally can be discretiz
ed with the help of a mixed
finite element method. Our approach uses a parametric finite e
lement method,
which can be shown to lead to good mesh properties. We prove st
ability estimates for
a semidiscrete (discrete in space, continuous in time) vers
ion of the method and show
existence and uniqueness results in the fully discrete case
. Finally, several numerical
results are presented involving convergence tests as well a
s the first computations
with Gaussian curvature and/or free or semi-free boundary c
onditions.
equations arising as
gradient flows for energy functionals that are quadratic in t
he principal curvatures
of a two-dimensional surface. Beside the well-known Willmo
re and Helfrich flows we
will also consider flows involving the Gaussian curvature of
the surface. Boundary
conditions for these flows are highly nonlinear, and we use a v
ariational approach to
derive weak formulations, which naturally can be discretiz
ed with the help of a mixed
finite element method. Our approach uses a parametric finite e
lement method,
which can be shown to lead to good mesh properties. We prove st
ability estimates for
a semidiscrete (discrete in space, continuous in time) vers
ion of the method and show
existence and uniqueness results in the fully discrete case
. Finally, several numerical
results are presented involving convergence tests as well a
s the first computations
with Gaussian curvature and/or free or semi-free boundary c
onditions.
Date Issued
2017-10-01
Date Acceptance
2017-01-09
Citation
IMA Journal of Numerical Analysis, 2017, 37 (4), pp.1657-1709
ISSN
0272-4979
Publisher
Oxford University Press (OUP)
Start Page
1657
End Page
1709
Journal / Book Title
IMA Journal of Numerical Analysis
Volume
37
Issue
4
Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record John W Barrett, Harald Garcke, Robert Nürnberg; Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature, IMA Journal of Numerical Analysis, Volume 37, Issue 4, 1 October 2017, Pages 1657–1709 is available online at: https://doi.org/10.1093/imanum/drx006
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Willmore flow
parametric finite elements
tangential movement
spontaneous curvature
clamped boundary conditions
Navier boundary conditions
Gaussian curvature energy
line energy
GEOMETRIC EVOLUTION-EQUATIONS
FINITE-ELEMENT-METHOD
FLUID MEMBRANES
LIPID-MEMBRANES
MEAN-CURVATURE
SURFACES
COMPUTATION
ALGORITHM
VESICLES
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
Numerical & Computational Mathematics
Publication Status
Published