Skip to main content

Preconditioning for Mixed Finite Element Formulations of Elliptic Problems

  • Conference paper
  • First Online:
Domain Decomposition Methods in Science and Engineering XX

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 91))

Summary

In this paper, we discuss a preconditioning technique for mixed finite element discretizations of elliptic equations. The technique is based on a block-diagonal approximation of the mass matrix which maintains the sparsity and positive definiteness of the corresponding Schur complement. This preconditioner arises from the multipoint flux mixed finite element method and is robust with respect to mesh size and is better conditioned for full permeability tensors than a preconditioner based on a diagonal approximation of the mass matrix.

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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

Access this chapter

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 PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

Bibliography

  1. M. B. Allen, R. E. Ewing, and P. Lu. Well-conditioned iterative schemes for mixed finite-element models of porous-media flows. SIAM J. Sci. Stat. Comput., 13:794–814, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numerica, 14:1–137, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Brezzi, J. Douglas, and L. D. Marini. Two families of mixed finite elements for second order elliptic problems. Numer. Math., 47(2):217–235, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. G. Ciarlet. The Finite Element Method for Elliptic Problems. Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978; reprinted, SIAM, Philadelphia, 2002.

    Google Scholar 

  5. R. E. Ewing, R. D. Lazarov, P. Lu, and P. Vassilevski. Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems. Lecture Notes in Mathematics, 1457:28–43, 1990.

    Article  MathSciNet  Google Scholar 

  6. R. Ingram, M. F. Wheeler, and I. Yotov. A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal., 48:1281–1312, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. F. Murphy, G. H. Golub, and A. J. Wathen. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput., 21:1969–1972, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. A. Raviart and J. Thomas. A mixed finite element method for 2-nd order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical aspects of the Finite Elements Method, Lectures Notes in Math. 606, pages 292–315. Springer, Berlin, 1977.

    Google Scholar 

  9. M. Wheeler, G. Xue, and I. Yotov. A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math., DOI 10.1007/s00211-011-0427-7, 2011.

    Google Scholar 

  10. M. F. Wheeler, G. Xue, and I. Yotov. A family of multipoint flux mixed finite element methods for elliptic problems on general grids. Procedia Computer Science, 4:918–927, 2011.

    Article  Google Scholar 

  11. M. F. Wheeler, G. Xue, and I. Yotov. A multipoint flux mixed finite element method on triangular prisms. Preprint, 2011.

    Google Scholar 

  12. M. F. Wheeler and I. Yotov. A multipoint flux mixed finite element method. SIAM. J. Numer. Anal., 44(5):2082–2106, 2006.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Guangri Xue is supported by Award No. KUS-F1-032-04, made by King Abdullah University of Science and Technology (KAUST).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tim Wildey .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Wildey, T., Xue, G. (2013). Preconditioning for Mixed Finite Element Formulations of Elliptic Problems. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_19

Download citation

Publish with us

Policies and ethics