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The Mimetic Finite Difference Method for Elliptic Problems

  • Book
  • © 2014

Overview

Authors:
  1. Lourenço Beirão Veiga

    1. Dipartimento di Matematica “Federico Enriques”, Università degli Studi di Milano, Italy

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  2. Konstantin Lipnikov

    1. Theoretical Division, Los Alamos National Laboratory, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Gianmarco Manzini

    1. Theoretical Division, Los Alamos National Laboratory, USA

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  • This is the first book on modern mimetic technology
  • The theoretical analysis is complemented by simple examples
  • The book covers a broad range of applications
  • Includes supplementary material: sn.pub/extras

Part of the book series: MS&A (MS&A, volume 11)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Foundation

    1. Front Matter

      Pages 1-1
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    2. Model elliptic problems

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 3-40

    3. Foundations of mimetic finite difference method

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 41-65

    4. Mimetic inner products and reconstruction operators

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 67-89

    5. Mimetic discretization of bilinear forms

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 91-113

  3. Mimetic Discretization of Basic PDEs

    1. Front Matter

      Pages 115-115
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    2. The diffusion problem in mixed form

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 117-154

    3. The diffusion problem in primal form

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 155-195

    4. Maxwell’s equations

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 197-219

    5. The Stokes problem

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 221-260

  4. Further Developments

    1. Front Matter

      Pages 261-261
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    2. Elasticity and plates

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 263-287

    3. Other linear and nonlinear mimetic schemes

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 289-310

    4. Analysis of parameters and maximum principles

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 311-337

    5. Diffusion problem on generalized polyhedral meshes

      • Lourenço Beirão da Veiga, Konstantin Lipnikov, Gianmarco Manzini
      Pages 339-370

  5. Back Matter

    Pages 371-394
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Keywords

About this book

This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications.

Reviews

From the book reviews:

“This book of about 400 pages is clear and relatively easy to read. It shows the capabilities and the efficiency of the mimetic finite difference method in the resolution of the usual partial differential equations, from their strong formulation. Many theoretical and practical aspects are addressed in detail. It is therefore highly recommended for anyone who wants to learn and use this method.” (Arnaud Münch, Mathematical Reviews, October, 2014)

“The research monograph is a useful source for scientists and engineers interested in computational treatment for various mathematical models arising in real life. It also proves to be a valuable research monograph for graduate students in Applied Mathematics or Computational Physics.” (Marius Ghergu, zbMATH, Vol. 1286, 2014)

Authors and Affiliations

  • Dipartimento di Matematica “Federico Enriques”, Università degli Studi di Milano, Italy

    Lourenço Beirão Veiga

  • Theoretical Division, Los Alamos National Laboratory, USA

    Konstantin Lipnikov, Gianmarco Manzini

About the authors

The Authors are active researchers in the field of numerical discretizations of partial differential equations. Together and in collaboration with other researchers, they published more than one-hundred papers on ISI ranked journals. Therefore, their expertise spans the formal mathematical analysis of the mimetic discretization methods, the application of this theoretical framework to real scientific and engineering models formulated in the setting of elliptic problems, and the computational properties of the numerical schemes discussed in the book.

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