Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Algebra in Scientific Computing

CASC 2015: Computer Algebra in Scientific Computing pp 376–390Cite as

Symbolic Computation and Finite Element Methods

(Invited Talk)

  • Conference paper
  • First Online:

  • 777 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9301))

Abstract

During the past few decades there have been many examples where computer algebra methods have been applied successfully in the analysis and construction of numerical schemes, including the computation of approximate solutions to partial differential equations. The methods range from Gröbner basis computations and Cylindrical Algebraic Decomposition to algorithms for symbolic summation and integration. The latter have been used to derive recurrence relations for efficient evaluation of high order finite element basis functions. In this paper we review some of these recent developments.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press (2000)

    Google Scholar 

  2. Becirovic, A., Paule, P., Pillwein, V., Riese, A., Schneider, C., Schöberl, J.: Hypergeometric Summation Algorithms for High Order Finite Elements. Computing 78(3), 235–249 (2006)

    Google Scholar 

  3. Beuchler, S., Pillwein, V.: Sparse shape functions for tetrahedral p-FEM using integrated Jacobi polynomials. Computing 80(4), 345–375 (2007)

    Google Scholar 

  4. Beuchler, S., Pillwein, V., Schöberl, J., Zaglmayr, S.: Sparsity optimized high order finite element functions on simplices. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing: Progress and Prospects. Texts and Monographs in Symbolic Computation, pp. 21–44. Springer, Wien (2012)

    Google Scholar 

  5. Beuchler, S., Pillwein, V., Zaglmay, S.: Fast summation techniques for sparse shape functions in tetrahedral hp-FEM. In: Domain Decomposition Methods in Science and Engineering XX, pp. 537–544 (2012)

    Google Scholar 

  6. Beuchler, S., Pillwein, V., Zaglmayr, S.: Sparsity optimized high order finite element functions for H(div) on simplices. Numerische Mathematik 122(2), 197–225 (2012)

    Google Scholar 

  7. Beuchler, S., Pillwein, V., Zaglmayr, S.: Sparsity optimized high order finite element functions for H(curl) on tetrahedra. Advances in Applied Mathematics 50, 749–769 (2013)

    Google Scholar 

  8. Beuchler, S., Schöberl, J.: New shape functions for triangular p-fem using integrated Jacobi polynomials. Numer. Math. 103, 339–366 (2006)

    Google Scholar 

  9. Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  10. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31, 333–390 (1977)

    Google Scholar 

  11. Brandt, A.: Rigorous Quantitative Analysis of Multigrid, I: Constant Coefficients Two-Level Cycle with L 2-Norm. SIAM J. on Numerical Analysis 31(6), 1695–1730 (1994)

    Google Scholar 

  12. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 2nd edn., vol. 15. Springer, New York (2002)

    Google Scholar 

  13. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)

    Google Scholar 

  14. Cluzeau, T., Dolean, V., Nataf, F., Quadrat, A.: Symbolic preconditioning techniques for linear systems of partial differential equations. In: Proceedings of 20th International Conference on Domain Decomposition Methods. accepted for publ. (2012)

    Google Scholar 

  15. Cluzeau, T., Dolean, V., Nataf, F., Quadrat, A.: Symbolic methods for developing new domain decomposition algorithms. Rapport de recherche RR-7953, INRIA, May 2012

    Google Scholar 

  16. Demkowicz, L.: Computing with hp Finite Elements I. One- and Two-Dimensional Elliptic and Maxwell Problems. CRC Press, Taylor and Francis (2006)

    Google Scholar 

  17. Hackbusch, W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)

    Google Scholar 

  18. Hong, H., Liska, R., Steinberg, S., Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. J. Symbolic Comput. 24(2), 161–187 (1997). Applications of quantifier elimination, Albuquerque, NM (1995)

    Google Scholar 

  19. Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for CFD. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, Oxford (1999)

    Google Scholar 

  20. Kauers, M.: SumCracker - A Package for Manipulating Symbolic Sums and Related Objects. J. Symbolic Comput. 41(9), 1039–1057 (2006)

    Google Scholar 

  21. Kauers, M.: Guessing Handbook. RISC Report Series 09–07, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria (2009)

    Google Scholar 

  22. Koutschan, Ch.: HolonomicFunctions (User’s Guide). Technical Report 10–01, RISC Report Series, University of Linz, Austria, January 2010

    Google Scholar 

  23. Koutschan, C., Lehrenfeld, C., Schöberl, J.: Computer algebra meets finite elements: an efficient implementation for maxwell’s equations. In: Numerical and Symbolic Scientific Computing: Progress and Prospects. Texts and Monographs in Symbolic Computation, vol. 1, pp. 105–121. Springer, Wien (2012)

    Google Scholar 

  24. Langer, U., Reitzinger, S., Schicho, J.: Symbolic methods for the element preconditioning technique. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 293–308. Springer, Heidelberg (2003)

    Google Scholar 

  25. Levandovskyy, V., Martin, B.: A symbolic approach to generation and analysis of finite difference schemes of partial differential equations. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing: Progress and Prospects. Texts & Monographs in Symbolic Computation, pp. 123–156. Springer, Wien (2011)

    Google Scholar 

  26. Pillwein, V., Takacs, S.: Smoothing analysis of an all-at-once multigrid approach for optimal control problems using symbolic computation. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing: Progress and Prospects. Springer, Wien (2011)

    Google Scholar 

  27. Pillwein, V., Takacs, S.: A local Fourier convergence analysis of a multigrid method using symbolic computation. J. Symbolic Comput. 63, 1–20 (2014)

    Google Scholar 

  28. Schwab, C.: p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (1998)

    Google Scholar 

  29. Szabó, B.A., Babuska, I: Finite Element Analysis. John Wiley & Sons (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RISC, Johannes Kepler University, Linz, Austria

    Veronika Pillwein

Authors
  1. Veronika Pillwein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Veronika Pillwein .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institute for Mathematics, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Hessen, Germany

    Werner M. Seiler

  4. Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Pillwein, V. (2015). Symbolic Computation and Finite Element Methods. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24021-3_28

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24020-6

  • Online ISBN: 978-3-319-24021-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation