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Numerical Analysis of Multiscale Problems pp 243–283Cite as

Mixed Multiscale Methods for Heterogeneous Elliptic Problems

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 83))

Abstract

We consider a second order elliptic problem written in mixed form, i.e., as a system of two first order equations. Such problems arise in many contexts, including flow in porous media. The coefficient in the elliptic problem (the permeability of the porous medium) is assumed to be spatially heterogeneous. The emphasis here is on accurate approximation of the solution with respect to the scale of variation in this coefficient. Homogenization and upscaling techniques alone are generally inadequate for this problem. As an alternative, multiscale numerical methods have been developed. They can be viewed in one of three equivalent frameworks: as a Galerkin or finite element method with nonpolynomial basis functions, as a variational multiscale method with standard finite elements, or as a domain decomposition method with restricted degrees of freedom on the interfaces. We treat each case, and discuss the advantages of the approach for devising effective local multiscale methods. Included is recent work on methods that incorporate information from homogenization theory and effective domain decomposition methods.

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Acknowledgements

This work was supported by the U.S. National Science Foundation and the Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001114.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX, 78712, USA

    Todd Arbogast

  2. Mathematics Department, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712–0257, USA

    Todd Arbogast

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  1. Todd Arbogast
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Correspondence to Todd Arbogast .

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Editors and Affiliations

  1. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Ivan G. Graham

  2. , Applied & Computational Mathematics, California Institute of Technology, Pasadena, 91125, California, USA

    Thomas Y. Hou

  3. , Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom

    Omar Lakkis

  4. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Robert Scheichl

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Arbogast, T. (2012). Mixed Multiscale Methods for Heterogeneous Elliptic Problems. In: Graham, I., Hou, T., Lakkis, O., Scheichl, R. (eds) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22061-6_8

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