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A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems

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Geometrically Unfitted Finite Element Methods and Applications

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 121))

Abstract

We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted background mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demonstrated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the background mesh.

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Notes

  1. 1.

    Quasi-uniformity is mainly assumed to simplify the overall presentation.

  2. 2.

    This is trivial since the mass term is already included in our form.

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Acknowledgements

This work was supported in part by the Kempe foundation (JCK-1612). The author expresses his gratitude to Ceren Gürkan for her help with the set-up of the convergence experiment, to Erik Burman for his great editorial assistance during the preparation of this contribution, and finally, to the two anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187, Umeå, Sweden

    André Massing

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  1. André Massing
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Correspondence to André Massing .

Editor information

Editors and Affiliations

  1. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, Luxembourg

    Stéphane P. A. Bordas

  2. Department of Mathematics, University College London, United Kingdom

    Erik Burman

  3. Department of Mathematics and Mathematical Statistics, Umeå University, Sweden

    Mats G. Larson

  4. Department of Mathematics, University of Houston, USA

    Maxim A. Olshanskii

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Massing, A. (2017). A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems. In: Bordas, S., Burman, E., Larson, M., Olshanskii, M. (eds) Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-71431-8_8

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