## AFEM for the Laplace-Beltrami operator on graphs: Design and conditional contraction property

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- by Khamron Mekchay, Pedro Morin and Ricardo H. Nochetto PDF
- Math. Comp.
**80**(2011), 625-648 Request permission

## Abstract:

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on $C^1$ graphs $\Gamma$ in $\mathbb {R}^d ~(d\ge 2)$. We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in $H^1(\Gamma )$ and the surface error in $W^1_\infty (\Gamma )$ due to approximation of $\Gamma$. We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of $\Gamma$ in $W^1_\infty$. We conclude with one numerical experiment that illustrates the theory.## References

- Mark Ainsworth and J. Tinsley Oden,
*A posteriori error estimation in finite element analysis*, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR**1885308**, DOI 10.1002/9781118032824 - Eberhard Bänsch, Pedro Morin, and Ricardo H. Nochetto,
*Surface diffusion of graphs: variational formulation, error analysis, and simulation*, SIAM J. Numer. Anal.**42**(2004), no. 2, 773–799. MR**2084235**, DOI 10.1137/S0036142902419272 - Susanne C. Brenner and L. Ridgway Scott,
*The mathematical theory of finite element methods*, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR**2373954**, DOI 10.1007/978-0-387-75934-0 - J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert,
*Quasi-optimal convergence rate for an adaptive finite element method*, SIAM J. Numer. Anal.**46**(2008), no. 5, 2524–2550. MR**2421046**, DOI 10.1137/07069047X - J. M. Cascón, K. Mekchay, P. Morin and R. H. Nochetto,
*Quasi-optimal convergence rate for AFEM for the Laplace-Beltrami operator on parametric surfaces*, (in preparation). - Zhiming Chen and Jia Feng,
*An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems*, Math. Comp.**73**(2004), no. 247, 1167–1193. MR**2047083**, DOI 10.1090/S0025-5718-04-01634-5 - Philippe G. Ciarlet,
*The finite element method for elliptic problems*, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR**0520174** - Ph. Clément,
*Approximation by finite element functions using local regularization*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér.**9**(1975), no. R-2, 77–84 (English, with Loose French summary). MR**0400739** - Alan Demlow,
*Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces*, SIAM J. Numer. Anal.**47**(2009), no. 2, 805–827. MR**2485433**, DOI 10.1137/070708135 - Alan Demlow and Gerhard Dziuk,
*An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces*, SIAM J. Numer. Anal.**45**(2007), no. 1, 421–442. MR**2285862**, DOI 10.1137/050642873 - Willy Dörfler,
*A convergent adaptive algorithm for Poisson’s equation*, SIAM J. Numer. Anal.**33**(1996), no. 3, 1106–1124. MR**1393904**, DOI 10.1137/0733054 - W. Dörfler and M. Rumpf,
*An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation*, Math. Comp.**67**(1998), no. 224, 1361–1382. MR**1489969**, DOI 10.1090/S0025-5718-98-00993-4 - Gerhard Dziuk,
*Finite elements for the Beltrami operator on arbitrary surfaces*, Partial differential equations and calculus of variations, Lecture Notes in Math., vol. 1357, Springer, Berlin, 1988, pp. 142–155. MR**976234**, DOI 10.1007/BFb0082865 - K. Mekchay,
*Convergence of adaptive finite element methods*, Ph.D. Dissertation, University of Maryland, December 2005. - Khamron Mekchay and Ricardo H. Nochetto,
*Convergence of adaptive finite element methods for general second order linear elliptic PDEs*, SIAM J. Numer. Anal.**43**(2005), no. 5, 1803–1827. MR**2192319**, DOI 10.1137/04060929X - Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert,
*Data oscillation and convergence of adaptive FEM*, SIAM J. Numer. Anal.**38**(2000), no. 2, 466–488. MR**1770058**, DOI 10.1137/S0036142999360044 - Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert,
*Convergence of adaptive finite element methods*, SIAM Rev.**44**(2002), no. 4, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic); MR1770058 (2001g:65157)]. MR**1980447**, DOI 10.1137/S0036144502409093 - Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert,
*Local problems on stars: a posteriori error estimators, convergence, and performance*, Math. Comp.**72**(2003), no. 243, 1067–1097. MR**1972728**, DOI 10.1090/S0025-5718-02-01463-1 - F. A. Ortega,
*GMV: General Mesh Viewer users Manual*, Version 1.8. Los Alamos National Laboratory, LAUR 95-2986, 1995. - Alfred Schmidt and Kunibert G. Siebert,
*Design of adaptive finite element software*, Lecture Notes in Computational Science and Engineering, vol. 42, Springer-Verlag, Berlin, 2005. The finite element toolbox ALBERTA; With 1 CD-ROM (Unix/Linux). MR**2127659** - R. Verfürth,
*A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Technique*, Wiley-Teubner, Chichester, 1996.

## Additional Information

**Khamron Mekchay**- Affiliation: Department of Mathematics, Faculty of Science, Chulalongkorn University, Phyathai, Bangkok 10330, Thailand – and – University of Maryland, College Park, Maryland 20742
- Email: k.mekchay@gmail.com
**Pedro Morin**- Affiliation: Instituto de Matemática Aplicada del Litoral, Universidad Nacional del Litoral, CONICET, Güemes 3450, S3000GLN Santa Fe, Argentina
- Email: pmorin@santafe-conicet.gov.ar
**Ricardo H. Nochetto**- Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
- MR Author ID: 131850
- Email: rhn@math.umd.edu
- Received by editor(s): July 20, 2009
- Received by editor(s) in revised form: February 26, 2010
- Published electronically: November 16, 2010
- Additional Notes: The first author was partially supported by NSF Grants DMS-0204670, DMS-0505454, and INT-0126272.

The second author was partially supported by CONICET through Grants PIP 5478, PIP 112-200801-02182, by Universidad Nacional del Litoral through Grants CAI+D 008-054 and CAI+D PI 062-312, and by NSF Grant DMS-0204670.

The third author was partially supported by NSF Grants DMS-0204670, DMS-0505454, DMS-0807811, and INT-0126272, and the General Research Board of the University of Maryland - © Copyright 2010 American Mathematical Society
- Journal: Math. Comp.
**80**(2011), 625-648 - MSC (2010): Primary 65N30, 65N50
- DOI: https://doi.org/10.1090/S0025-5718-2010-02435-4
- MathSciNet review: 2772090