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AFEM for Geometric PDE: The Laplace-Beltrami Operator

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Analysis and Numerics of Partial Differential Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 4))

Abstract

We present several applications governed by geometric PDE, and their parametric finite element discretization, which might yield singular behavior. The success of such discretization hinges on an adequate variational formulation of the Laplace-Beltrami operator, which we describe in detail for polynomial degree 1. We next present a complete a posteriori error analysis which accounts for the usual PDE error as well as the geometric error induced by interpolation of the surface. This leads to an adaptive finite element method (AFEM) and its convergence. We discuss a contraction property of AFEM and show its quasi-optimal cardinality.

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Acknowledgements

The work of A. Bonito was partially supported by NSF Grant DMS-0914977.

The work of J.M. Cascón was partially supported by Secretaría de Estado de Investigación, Desarrollo e Innovación through grant: CGL2011-29396-C03-02 (Spain), and by Conserjería de Educación (Junta de Castilla y León), through grant: SA266A12-2.

The work of P. Morin was partially supported by CONICET through grant PIP 112-200801-02182, Universidad Nacional del Litoral through grants CAI+D 062-312, 062-309, and Agencia Nacional de Promoción Científica y Tecnológica, through grant PICT-2008-0622 (Argentina).

The work of R.H. Nochetto was partially supported by NSF Grant DMS-1109325, and the General Research Board of the University of Maryland.

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX, 77843-3368, USA

    Andrea Bonito

  2. Departamento de Economía e Historia Económica, Universidad de Salamanca, Salamanca, 37008, Spain

    J. Manuel Cascón

  3. Instituto de Matemática Aplicada del Litoral (IMAL), Güemes 3450 and Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, 3000, Santa Fe, Argentina

    Pedro Morin

  4. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

Authors
  1. Andrea Bonito
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  2. J. Manuel Cascón
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  3. Pedro Morin
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  4. Ricardo H. Nochetto
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Corresponding author

Correspondence to Ricardo H. Nochetto .

Editor information

Editors and Affiliations

  1. Ist. Matematica Applicata e, Tecnologie Informatiche (IMATI), CNR Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Franco Brezzi

  2. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Piero Colli Franzone

  3. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Ugo Gianazza

  4. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Gianni Gilardi

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In memory of Enrico Magenes.

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Bonito, A., Cascón, J.M., Morin, P., Nochetto, R.H. (2013). AFEM for Geometric PDE: The Laplace-Beltrami Operator. In: Brezzi, F., Colli Franzone, P., Gianazza, U., Gilardi, G. (eds) Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano. https://doi.org/10.1007/978-88-470-2592-9_15

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