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A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces


Authors: Maxim A. Olshanskii and Danil Safin
Journal: Math. Comp. 85 (2016), 1549-1570
MSC (2010): Primary 65N15, 65N30, 76D45, 76T99
DOI: https://doi.org/10.1090/mcom/3030
Published electronically: September 16, 2015
MathSciNet review: 3471100
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Abstract: This paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $ \mathbb{R}^N$, $ N=2,3$. The method allows a surface to be given implicitly as a zero level of a level set function. A surface equation is extended to a narrow-band neighborhood of the surface. The resulting extended equation is a non-degenerate PDE, and it is solved on a bulk mesh that is unaligned to the surface. An unfitted finite element method is used to discretize extended equations. Error estimates are proved for finite element solutions in the bulk domain and restricted to the surface. The analysis admits finite elements of a higher order and gives sufficient conditions for archiving the optimal convergence order in the energy norm. Several numerical examples illustrate the properties of the method.


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Additional Information

Maxim A. Olshanskii
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: molshan@math.uh.edu

Danil Safin
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: dksafin@math.uh.edu

DOI: https://doi.org/10.1090/mcom/3030
Keywords: Surface, PDE, finite element method, level set method, error analysis
Received by editor(s): January 29, 2014
Received by editor(s) in revised form: December 30, 2014
Published electronically: September 16, 2015
Additional Notes: This work has been supported by NSF through the Division of Mathematical Sciences grant 1315993
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society