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Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

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Title: Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature
Authors: Barrett, JW
Garcke, H
Nurnberg, R
Item 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.
Issue Date: 1-Oct-2017
Date of Acceptance: 9-Jan-2017
URI: http://hdl.handle.net/10044/1/43913
DOI: https://dx.doi.org/10.1093/imanum/drx006
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
Keywords: 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
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences