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Principles of Mimetic Discretizations of Differential Operators

  • Conference paper
Compatible Spatial Discretizations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 142))

Abstract

Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all attendant discrete structures are put together by using two basic mappings between differential forms and cochains. The key concept of the framework is a natural inner product on cochains which induces a combinatorial Hodge theory on the cochain complex. The framework supports mutually consistent operations of differentiation and integration, has a combinatorial Stokes theorem, and preserves the invariants of the De Rham cohomology groups. This allows, among other things, for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume, and finite difference methods. We describe how these methods result from a choice of the reconstruction operator and explain when they are equivalent. We demonstrate how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian.

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Author information

Authors and Affiliations

  1. Computational Mathematics and Algorithms, Sandia National Laboratories, Mail Stop 1110, Albuquerque, NM, 87185

    Pavel B. Bochev

  2. Mathematical Modeling and Analysis, Los Alamos National Laboratory, T-7 Mail Stop B284, Los Alamos, NM, 87545

    James M. Hyman

Authors
  1. Pavel B. Bochev
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  2. James M. Hyman
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Editor information

Editors and Affiliations

  1. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, 55455, USA

    Douglas N. Arnold

  2. Computational Mathematics and Algorithms Department, Sandia National Laboratories, Alburquerque, NM, 87185-1110, USA

    Pavel B. Bochev  & Richard B. Lehoucq  & 

  3. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213-3890, USA

    Roy A. Nicolaides

  4. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Mikhail Shashkov

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Bochev, P.B., Hyman, J.M. (2006). Principles of Mimetic Discretizations of Differential Operators. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol 142. Springer, New York, NY. https://doi.org/10.1007/0-387-38034-5_5

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