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Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature
Barrett, John W., Garcke, Harald and Nürnberg, Robert (2016) Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature. Preprintreihe der Fakultät Mathematik 1/2016, Working Paper. (Submitted)Date of publication of this fulltext: 21 Jun 2016 12:35
Monograph
DOI to cite this document: 10.5283/epub.33896
Abstract
We study numerical approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in the principal curvatures of a two-dimensional surface. Beside the well-known Willmore 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 ...
We study numerical approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in the principal curvatures of a two-dimensional surface. Beside the well-known Willmore 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 variational approach to derive weak formulations, which naturally can be discretized with the help of a mixed finite element method. Our approach uses a parametric finite element method,
which can be shown to lead to good mesh properties. We prove stability estimates for a semidiscrete (discrete in space, continuous in time) version 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 as the first computations with Gaussian curvature and/or free or semi-free boundary conditions.
Involved Institutions
Details
| Item type | Monograph (Working Paper) | ||||||||||
| Series of the University of Regensburg: | Preprintreihe der Fakultät Mathematik | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Volume: | 1/2016 | ||||||||||
| Date | 2016 | ||||||||||
| Institutions | Mathematics > Prof. Dr. Harald Garcke | ||||||||||
| Classification |
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| Keywords | Willmore flow, parametric finite elements, tangential movement, spontaneous curvature, clamped boundary conditions, Navier boundary conditions, Gaussian curvature energy, line energy | ||||||||||
| Dewey Decimal Classification | 500 Science > 510 Mathematics | ||||||||||
| Status | Submitted | ||||||||||
| Refereed | No, this version has not been refereed yet (as with preprints) | ||||||||||
| Created at the University of Regensburg | Yes | ||||||||||
| URN of the UB Regensburg | urn:nbn:de:bvb:355-epub-338961 | ||||||||||
| Item ID | 33896 |
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