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A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces


Authors: Karl Larsson and Mats G. Larson
Journal: Math. Comp.
MSC (2010): Primary 65N15, 65N30, 58J99
DOI: https://doi.org/10.1090/mcom/3179
Published electronically: March 30, 2017
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Abstract: We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in $ \mathbb{R}^3$. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in $ L^2$ norm. This can be seen as an extension of the formalism and method originally used by Dziuk (1988) for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation $ \Gamma _h$ of an implicitly defined surface $ \Gamma $ we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on $ \Gamma $. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.


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Additional Information

Karl Larsson
Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden
Email: karl.larsson@umu.se

Mats G. Larson
Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden
Email: mats.larson@umu.se

DOI: https://doi.org/10.1090/mcom/3179
Received by editor(s): January 19, 2015
Received by editor(s) in revised form: February 5, 2016, and April 26, 2016
Published electronically: March 30, 2017
Additional Notes: This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grant No. 2013-4708, and the Swedish strategic research programme eSSENCE
Article copyright: © Copyright 2017 American Mathematical Society