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Monotone Multigrid Methods Based on Parametric Finite Elements

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Domain Decomposition Methods in Science and Engineering XX

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 91))

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Summary

In this paper, a particular technique for the application of elementary multilevel ideas to problems with warped boundaries is studied in the context of the numerical simulation of elastic contact problems. Combining a general multilevel setting with a different perspective, namely an advanced geometric modeling point of view, we present a (monotone) multigrid method based on a hierarchy of parametric finite element spaces. For the construction, a full-dimensional parameterization of high order is employed which accurately represents the computational domain.The purpose of the volume parametric finite element discretization put forward here is two-fold. On the one hand, it allows for an elegant multilevel hierarchy to be used in preconditioners. On the other hand, it comes with particular advantages for the modeling of contact problems. After all, the long-term objective lies in an increased flexibility of h p-adaptive methods for contact problems.

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Acknowledgements

The authors would like to thank Helmut Harbrecht and Maharavo Randrianarivony for bringing this topic to their attention. Moreover, we acknowledge the latter for providing his code for the tetrahedral transfinite interpolation described in [15]. The valuable assistance of Lukas Döring in the implementation of a flexible interface of the parameterization concept to our finite element code is appreciated. This work was supported by the Bonn International Graduate School in Mathematics and the Ford University Research Program.

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Authors and Affiliations

  1. Institute of Computational Science, University of Lugano, Via G. Buffi 13, 6904, Lugano, Switzerland

    Thomas Dickopf & Rolf Krause

Authors
  1. Thomas Dickopf
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  2. Rolf Krause
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Correspondence to Thomas Dickopf .

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Editors and Affiliations

  1. , Department of Mathematics, University of California, San Diego, Gilman Drive 9500, La Jolla, 92093-0112, California, USA

    Randolph Bank

  2. , Department of Mathematics, University of California, San Diego, Gilman Drive 9500, La Jolla, 92093-0112, California, USA

    Michael Holst

  3. , Courant Institute of Mathematical, New York University, Mercer Street 251, New York, 10012-1185, New York, USA

    Olof Widlund

  4. , Department of Mathematics, Pennsylvania State University, State College, 16802, Pennsylvania, USA

    Jinchao Xu

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Dickopf, T., Krause, R. (2013). Monotone Multigrid Methods Based on Parametric Finite Elements. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_37

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