Skip to main content

An Algebraic Multigrid Method for an Adaptive Space–Time Finite Element Discretization

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10665)

Abstract

This work is devoted to numerical studies on an algebraic multigrid preconditioned GMRES method for solving the linear algebraic equations arising from a space–time finite element discretization of the heat equation using h–adaptivity on tetrahedral meshes. The finite element discretization is based on a Galerkin–Petrov variational formulation using piecewise linear finite elements simultaneously in space and time. In this work, we focus on h–adaptivity relying on a residual based a posteriori error estimation, and study some important components in the algebraic multigrid method for solving the space–time finite element equations.

Keywords

  • Adaptive space–time finite element
  • Algebraic multigrid

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-73441-5_6
  • Chapter length: 8 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-73441-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.00
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

References

  1. Andreev, R.: Wavelet-in-time multigrid-in-space preconditioning of parabolic evolution equations. SIAM J. Sci. Comput. 38(1), A216–A242 (2016)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Arnold, D.N., Mukherjee, A., Pouly, L.: Locally adapted tetrahedral meshes using bisection. SIAM J. Sci. Comput. 22(2), 431–448 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Bank, R.E., Vassilevski, P.S., Zikatanov, L.T.: Arbitrary dimension convection-diffusion schemes for space-time discretizations. J. Comput. Appl. Math. 310, 19–31 (2017)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Bey, J.: Tetrahedral grid refinement. Computing 55(4), 355–378 (1995)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Briggs, W.L., Henson, V.E., McCormick, S.F.: A multigrid tutorial. SIAM, Philadelphia (2000)

    CrossRef  MATH  Google Scholar 

  6. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Ellis, T.E.: Space-time discontinuous Petrov-Galerkin finite elements for transient fluid mechanics. Ph.D. thesis. University of Texas at Austin (2016)

    Google Scholar 

  8. Ellis, T.E., Demkowicz, L., Chan, J.: Locally conservative discontinuous Petrov-Galerkin finite elements for fluid problems. Comput. Math. Appl. 68(11), 1530–1549 (2014)

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Gander, M.J., Neumüller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38(4), A2173–A2208 (2016)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Hughes, T.J.R., Hulbert, G.M.: Space-time finite element methods for elastodynamics: formulations and error estimates. Comput. Methods Appl. Math. 66(3), 339–363 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Langer, U., Moore, S.E., Neumüller, M.: Space-time isogeometric analysis of parabolic evolution problems. Comput. Methods Appl. Math. 306, 342–363 (2016)

    MathSciNet  Google Scholar 

  13. MacLachlan, S., Saad, Y.: A greedy strategy for coarse-grid selection. SIAM J. Sci. Comput. 29(5), 1825–1853 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  14. Moore, P.K.: A posteriori error estimation with finite element semi-and fully discrete methods for nonlinear parabolic equations in one space dimension. SIAM J. Numer. Anal. 31(1), 149–169 (1994)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Neumüller, M.: Space-time methods: fast solvers and applications. Ph.D. thesis. TU Graz (2013)

    Google Scholar 

  16. Neumüller, M., Steinbach, O.: Refinement of flexible space-time finite element meshes and discontinuous Galerkin methods. Comput. Vis. Sci. 14, 189–205 (2011)

    MathSciNet  CrossRef  Google Scholar 

  17. Popa, C.: Algebraic multigrid smoothing property of Kaczmarz’s relaxation for general rectangular linear systems. Electron. Trans. Numer. Anal. 29, 150–162 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Schmich, M., Vexler, B.: Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369–393 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  19. Schwab, C., Stevenson, R.: Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comput. 78(267), 1293–1318 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. Steinbach, O.: Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15, 551–566 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Steinbach, O., Yang, H.: An adaptive space-time finite element method for solving the heat equation, Technical report. TU Graz (2017, in preparation)

    Google Scholar 

  22. Urban, K., Patera, A.T.: An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83(288), 1599–1615 (2014)

    MathSciNet  CrossRef  MATH  Google Scholar 

  23. Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford Unversity Press, Oxford (2013)

    CrossRef  MATH  Google Scholar 

  24. Zhang, S.: Multi-level iterative techniques. Ph.D. thesis. Penn State University (1988)

    Google Scholar 

Download references

Acknowledgements

This work has been supported by the Austrian Science Fund (FWF) under the Grant SFB Mathematical Optimisation and Applications in Biomedical Sciences.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Olaf Steinbach .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Steinbach, O., Yang, H. (2018). An Algebraic Multigrid Method for an Adaptive Space–Time Finite Element Discretization. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-73441-5_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73440-8

  • Online ISBN: 978-3-319-73441-5

  • eBook Packages: Computer ScienceComputer Science (R0)