Skip to main content
SpringerLink
Log in

Preconditioned Space-Time Boundary Element Methods for the One-Dimensional Heat Equation

  • Conference paper
  • First Online:
Domain Decomposition Methods in Science and Engineering XXIV (DD 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 125))

Included in the following conference series:

  • 707 Accesses

Abstract

In this note we describe a space-time boundary element discretization of the heat equation and an efficient and robust preconditioning strategy which is based on the use of boundary integral operators of opposite orders, but which requires a suitable stability condition for the boundary element spaces used for the discretization. We demonstrate the method for the simple spatially one-dimensional case. However, the presented results, particularly the stability analysis of the boundary element spaces, can be used to extend the method to the two- and three-dimensional problem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Costabel, Boundary integral operators for the heat equation. Integr. Equ. Oper. Theory 13, 498–552 (1990)

    Article  MathSciNet  Google Scholar 

  2. S. Dohr, K. Niino, O. Steinbach, Preconditioned space-time boundary element methods for the heat equation (in preparation)

    Google Scholar 

  3. R. Hiptmair, Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006)

    Article  MathSciNet  Google Scholar 

  4. U. Langer, O. Steinbach, Boundary element tearing and interconnecting methods. Computing 71, 205–228 (2003)

    Article  MathSciNet  Google Scholar 

  5. J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications II (Springer, Berlin, 1972)

    MATH  Google Scholar 

  6. P.J. Noon, The single layer heat potential and Galerkin boundary element methods for the heat equation. Ph.D. Thesis, University of Maryland, 1988

    Google Scholar 

  7. O. Steinbach, On the stability of the L 2 projection in fractional Sobolev spaces. Numer. Math. 88, 367–379 (2001)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. O. Steinbach, W.L. Wendland, Efficient preconditioners for boundary element methods and their use in domain decomposition, in Domain Decomposition Methods in Sciences in Engineering: 8th International Conference, Beijing, China, ed. by R. Glowinski et al. (Wiley, Hoboken, 1997), pp. 3–18

    Google Scholar 

  10. O. Steinbach, W.L. Wendland, The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9, 191–216 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the International Research Training Group 1754, funded by the German Research Foundation (DFG) and the Austrian Science Fund (FWF). Additionally, S. Dohr would like to acknowledge the financial support provided by the University of Bergen.

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, TU Graz, Graz, Austria

    Stefan Dohr & Olaf Steinbach

Authors
  1. Stefan Dohr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olaf Steinbach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Stefan Dohr or Olaf Steinbach .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Petter E. Bjørstad

  2. Department of Mathematics, Louisana State University, Baton Rouge, LA, USA

    Susanne C. Brenner

  3. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Lawrence Halpern

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Korea (Republic of)

    Hyea Hyun Kim

  5. FB Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Ralf Kornhuber

  6. Western Norway University of Applied Sciences, Bergen, Norway

    Talal Rahman

  7. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Olof B. Widlund

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dohr, S., Steinbach, O. (2018). Preconditioned Space-Time Boundary Element Methods for the One-Dimensional Heat Equation. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_22

Download citation

Publish with us

Policies and ethics

Search

Navigation