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Mixed Methods for Elliptic Problems

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Mixed Finite Element Methods and Applications

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 44))

Abstract

This chapter will present a first set of applications of the theory developed in the previous chapters. It will provide us with the occasion of introducing many ideas which often have a more general scope than the simple considered case. We shall indeed consider the most simple cases of non-standard methods for Dirichlet’s problem, including hybrid methods. We then concentrate on numerical issues for the solution of the discrete problems arising from the previous constructions. In the following section, we sketch miscellaneous results on error estimates in different norms. Section 7.6 is dedicated to an example of application to semiconductor devices simulation. Section 7.7 discusses the sensitivity of low order mixed formulations to mesh deformation. We shall then consider in Sect. 7.8 the relations between mixed methods and the Finite Volume Method. A related idea, using a nonconforming element, will then be discussed in Sect. 7.9 and shown not to be convergent. Finally, Sect. 7.10 presents some applications of augmented formulations introduced in Sect. 1.5.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, University of Pavia, Pavia, Italy

    Daniele Boffi

  2. IUSS (Istituto Universitario di Studi Superiori), Pavia, Italy

    Franco Brezzi

  3. Département de mathématiques et de statistique, Université Laval, Québec, Canada

    Michel Fortin

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  1. Daniele Boffi
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  2. Franco Brezzi
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  3. Michel Fortin
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Boffi, D., Brezzi, F., Fortin, M. (2013). Mixed Methods for Elliptic Problems. In: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36519-5_7

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