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An Alternative to the Least-Squares Mixed Finite Element Method for Elliptic Problems

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Numerical Mathematics and Advanced Applications

Summary

In this paper we derive a strengthened Cauchy-Schwarz inequality that enables us to formulate a short and transparant proof of the coercivity of a Least Squares Mixed Finite Element bilinear form. Also, it shows that the coupling between H 10 (Ω) and H(div; Ω) is weak enough to be neglected. This results in an alternative way to compute approximations of both the scalar variable and its gradient for second order elliptic problems.

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References

  1. Arnold, D.N., Falk, R.S, Winther, R. (1997). Preconditioning in H(div; Ω) and applications. Math. Comp., 66, 957–984.

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  2. Brandts, J.H., Chen, Y., Yang, J. (2003). Analysis of least-squares mixed finite elements in terms of standard and mixed elements. UvA Numerica Preprint 08, University of Amsterdam, Netherlands. Submitted.

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  3. Pehlivanov, A.I., Carey, G.F., Lazarov, R.D. (1994). Least-squares Mixed Finite Elements for Second Order Elliptic Problems. SIAM J. Numer. Anal., 31, 1368–1377.

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  4. Pehlivanov, A.I., Carey, G.F., Vassilevski, P.S. (1996): Least-squares mixed finite element methods for non-selfadjoint elliptic problems. I. Error estimates. Numer. Math., 72, 501–522.

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Author information

Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, Faculty of Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, Netherlands

    Jan Brandts

  2. Department of Mathematics, Xiangtan University, Xiangtan, 411105, China

    Yanping Chen

Authors
  1. Jan Brandts
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  2. Yanping Chen
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Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Physics Department of Numerical Mathematics, Charles University Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Miloslav Feistauer , Vít Dolejší , Petr Knobloch  & Karel Najzar , ,  & 

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© 2004 Springer-Verlag Berlin Heidelberg

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Brandts, J., Chen, Y. (2004). An Alternative to the Least-Squares Mixed Finite Element Method for Elliptic Problems. In: Feistauer, M., Dolejší, V., Knobloch, P., Najzar, K. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18775-9_14

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  • DOI: https://doi.org/10.1007/978-3-642-18775-9_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62288-5

  • Online ISBN: 978-3-642-18775-9

  • eBook Packages: Springer Book Archive

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