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Reparameterization and Adaptive Quadrature for the Isogeometric Discontinuous Galerkin Method

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Mathematical Methods for Curves and Surfaces (MMCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10521))

Abstract

We use the Poisson problem with Dirichlet boundary conditions to illustrate the complications that arise from using non-matching interface parameterizations within the framework of Isogeometric Analysis on a multi-patch domain, using discontinuous Galerkin (dG) techniques to couple terms across the interfaces. The dG-based discretization of a partial differential equation is based on a modified variational form, where one introduces additional terms that measure the discontinuity of the values and normal derivatives across the interfaces between patches. Without matching interface parameterizations, firstly, one needs to identify pairs of associated points on the common interface of the two patches for correctly evaluating the additional terms. We will use reparameterizations to perform this task. Secondly, suitable techniques for numerical integration are needed to evaluate the quantities that occur in the discretization with the required level of accuracy. We explore two possible approaches, which are based on subdivision and adaptive refinement, respectively.

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Notes

  1. 1.

    For simplicity we consider uniform knots only. If this is not the case then one may consider quasi-uniform knots instead, choosing a parameter that controls the size of all knot spans.

  2. 2.

    G+Smo: gs.jku.at.

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Authors and Affiliations

  1. Doctoral Program “Computational Mathematics”, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Agnes Seiler

  2. Institute of Applied Geometry, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Bert Jüttler

Authors
  1. Agnes Seiler
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  2. Bert Jüttler
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Corresponding author

Correspondence to Agnes Seiler .

Editor information

Editors and Affiliations

  1. University of Oslo, Oslo, Norway

    Michael Floater

  2. University of Oslo, Oslo, Norway

    Tom Lyche

  3. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  4. University of Oslo, Oslo, Norway

    Knut Mørken

  5. Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Seiler, A., Jüttler, B. (2017). Reparameterization and Adaptive Quadrature for the Isogeometric Discontinuous Galerkin Method. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science(), vol 10521. Springer, Cham. https://doi.org/10.1007/978-3-319-67885-6_14

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  • DOI: https://doi.org/10.1007/978-3-319-67885-6_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67884-9

  • Online ISBN: 978-3-319-67885-6

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