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A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary

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Abstract

We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.

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Authors and Affiliations

  1. Department of Mathematics, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA, 94720-3840, USA

    August Johansson

  2. Department of Mathematics, Umeå University, 901 87 , Umeå, Sweden

    Mats G. Larson

Authors
  1. August Johansson
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  2. Mats G. Larson
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Correspondence to August Johansson.

Additional information

A. Johansson is supported by the Miller Institute for Basic Research in Science.

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Johansson, A., Larson, M.G. A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123, 607–628 (2013). https://doi.org/10.1007/s00211-012-0497-1

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  • DOI: https://doi.org/10.1007/s00211-012-0497-1

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