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Deckelnick, K, Elliott, CM and Ranner, T (2014) Unfitted finite element methods using bulk meshes for surface partial differential equations. SIAM Journal on Numerical Analysis, 52 (4). 2137 - 2162. ISSN 0036-1429
Abstract
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ⊂Rn+1, is embedded in a polyhedral domain in Rn+1 consisting of a union, Th, of (n+1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on Th. Our first method is a sharp interface method, \emph{SIF}, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γh, of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes \emph{SIF} from the method of [42]. The second method, \emph{NBM}, is a narrow band method in which the region of integration is a narrow band of width O(h). \emph{NBM} is similar to the method of [13]. but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L2 and H1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2014 Society for Industrial and Applied Mathematics. Reproduced in accordance with the publisher's self-archiving policy. |
Keywords: | Unfitted finite elements; cut cells; error analysis; narrow band; sharp interface; elliptic and parabolic surface equations |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Institute for Computational and Systems Science (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 23 Oct 2014 09:04 |
Last Modified: | 23 Jun 2023 21:41 |
Published Version: | http://dx.doi.org/10.1137/130948641 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Identification Number: | 10.1137/130948641 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:80492 |
Available Versions of this Item
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Unfitted finite element methods using bulk meshes for surface partial differential equations. (deposited 22 Jan 2014 11:48)
- Unfitted finite element methods using bulk meshes for surface partial differential equations. (deposited 23 Oct 2014 09:04) [Currently Displayed]