Skip to main content
SpringerLink
Log in

Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

This paper is concerned with using discontinuous Galerkin isogeometric analysis (dG-IGA) as a numerical treatment of diffusion problems on orientable surfaces \({\Omega } \subset \mathbb {R}^{3}\). The computational domain or surface considered consists of several non-overlapping subdomains or patches which are coupled via an interior penalty scheme. In Langer and Moore [13], we presented a priori error estimate for conforming computational domains with matching meshes across patch interface and a constant diffusion coefficient. However, in this article, we generalize the a priori error estimate to non-matching meshes and discontinuous diffusion coefficients across patch interfaces commonly occurring in industry. We construct B-spline or NURBS approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate for the symmetric discontinuous Galerkin scheme and numerical experiments to confirm the theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics 140, Elsevier/Academic Press, second edition (2008)

  2. Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Comput. Methods Appl. Mech. Engrg. 194, 4135–4195 (2006)

    MATH  Google Scholar 

  3. Beirão da Veiga, L., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numerica 23(5), 157–287 (2014)

    Article  MathSciNet  Google Scholar 

  4. Dede, L., Quarteroni, A.: Isogeometric analysis for second order partial differential equations on surfaces. Comput. Methods Appl. Mech Engrg. 284, 807–834 (2015)

    Article  MathSciNet  Google Scholar 

  5. Dedner, A., Madhavan, P., Stinner, B.: Analysis of the discontinuous Galerkin method for elliptic problems on surfaces. IMA J. Numer Anal. 33(3), 952–973 (2013)

    Article  MathSciNet  Google Scholar 

  6. Di Pietro, D. A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, volume 69 of Mathématiques et Applications. Springer, Heidelberg (2012)

    Google Scholar 

  7. Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numerica 22, 289–396 (2013)

    Article  MathSciNet  Google Scholar 

  8. Evans, J.A., Hughes, T.J.R.: Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Numer. Math. 123(2), 259–290 (2013)

    Article  MathSciNet  Google Scholar 

  9. Hofer, C.: Analysis of discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods. Math. Models Methods Appl. Sci. 28(01), 131–158 (2018)

    Article  MathSciNet  Google Scholar 

  10. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg. 194, 4135–4195 (2005)

    Article  MathSciNet  Google Scholar 

  11. Jüttler, B., Langer, U., Mantzaflaris, A., Moore, S.E., Zulehner, W.: Geometry + simulation modules: implementing isogeometric analysis. PAMM 14(1), 961–962 (2014)

    Article  Google Scholar 

  12. Langer, U., Mantzaflaris, A., Moore, S.E., Toulopoulos, I.: Multipatch discontinuous Galerkin isogeometric analysis. In: Jüttler, B., Simeon, B. (eds.) Isogeometric Analysis and Applications 2014, volume 107 of Lecture Notes in Computational Science and Engineering, pp 1–32. Springer (2015)

  13. Langer, U., Moore, S.E.: Domain decomposition methods in science and engineering xxii. In: Dickopf, T., Gander, J.M., Halpern, L., Krause, R., Luca Pavarino, F. (eds.) Domain Decomposition Methods in Science and Engineering XXII, chapter Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces, pp 319–326. Springer, Cham (2016)

  14. Moore, S.E.: Nonstandard Discretization Strategies In Isogeometric Analysis for Partial Differential Equations. PhD thesis, Johannes Kepler University (2017)

  15. Moore, S.E.: Discontinuous Galerkin isogeometric analysis for the biharmonic equation. Comput. Math. Appl. 76(4), 673–685 (2018)

    Article  MathSciNet  Google Scholar 

  16. Shahbazi, K.: An explicit expression for the penalty parameter of the interior penalty method. J. Comput. Phys. 205(2), 401–407 (2005)

    Article  Google Scholar 

  17. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  18. Wu, M., Wang, Y., Mourrain, B., Nkonga, B, Cheng, C.: Convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis. Comput. Aided Geometric Des. 52-53, 170–189 (2017). 2017 Geometric Modeling and Processing

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author acknowledges the Horizon 2020 Programme (2014–2020) under grant agreement number 678727.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Cape Coast, Cape Coast, Ghana

    Stephen Edward Moore

Authors
  1. Stephen Edward Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen Edward Moore.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moore, S.E. Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces. Numer Algor 85, 1075–1094 (2020). https://doi.org/10.1007/s11075-019-00856-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00856-x

Keywords

Search

Navigation