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Discretization of Mixed Formulations of Elliptic Problems on Polyhedral Meshes

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Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 114))

Abstract

We review basic design principles underpinning the construction of the mimetic finite difference and a few finite volume and finite element schemes for mixed formulations of elliptic problems. For a class of low-order mixed-hybrid schemes, we show connections between these principles and prove that the consistency and stability conditions must lead to a member of the mimetic family of schemes regardless of the selected discretization framework. Finally, we give two examples of using flexibility of the mimetic framework: derivation of arbitrary-order schemes and inexpensive convergent schemes for nonlinear problems with small diffusion coefficients.

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Acknowledgements

This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The authors acknowledge the support of the US Department of Energy Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research.

The meshes for the Marshak problem were created and managed using the mesh generation toolset MSTK (software.lanl.gov/MeshTools/trac) developed by Dr. Rao Garimella at Los Alamos National Laboratory.

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Authors and Affiliations

  1. Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Konstantin Lipnikov & Gianmarco Manzini

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  1. Konstantin Lipnikov
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  2. Gianmarco Manzini
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Correspondence to Konstantin Lipnikov .

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Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Strathclyde, Glasgow, United Kingdom

    Gabriel R. Barrenechea

  2. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  3. Department of Mathematics, University of Leicester, Leicester, United Kingdom

    Andrea Cangiani

  4. Department of Mathematics, National Technical University of Athens, Zografou, Greece

    Emmanuil H. Georgoulis

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Lipnikov, K., Manzini, G. (2016). Discretization of Mixed Formulations of Elliptic Problems on Polyhedral Meshes. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41640-3_10

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