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Numerical Methods and Analysis of Multiscale Problems

  • Book
  • © 2017

Overview

Authors:
  1. Alexandre L. Madureira

    1. Laboratório Nacional de Computação Científica-LNCC, Petrópolis, Brazil

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  • Offers an introduction to asymptotic analysis techniques and various finite element methods for elliptic problems
  • Presents numerous case studies on modeling techniques of multiscale PDEs, in one- and two-dimensional domains
  • Explores modeling through both continuous and numerical approximations
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-x
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  2. Introductory Material and Finite Element Methods

    • Alexandre L. Madureira
    Pages 1-21

  3. One-Dimensional Singular Perturbed Problems

    • Alexandre L. Madureira
    Pages 23-38

  4. An Application in Neuroscience: Heterogeneous Cable Equation

    • Alexandre L. Madureira
    Pages 39-47

  5. Two-Dimensional Reaction-Diffusion Equations

    • Alexandre L. Madureira
    Pages 49-65

  6. Modeling PDEs in Domains with Rough Boundaries

    • Alexandre L. Madureira
    Pages 67-84

  7. Partial Differential Equations with Oscillatory Coefficients

    • Alexandre L. Madureira
    Pages 85-107

  8. Back Matter

    Pages 109-123
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Keywords

About this book

This book is about numerical modeling of multiscale problems, and introduces several asymptotic analysis and numerical techniques which are necessary for a proper approximation of equations that depend on different physical scales.  Aimed at advanced undergraduate and graduate students in mathematics, engineering and physics – or researchers seeking a no-nonsense approach –, it discusses  examples in their simplest possible settings, removing mathematical hurdles that might hinder a clear understanding of the methods.

The problems considered are given by singular perturbed reaction advection diffusion equations in one and two-dimensional domains, partial differential equations in domains with rough boundaries, and equations with oscillatory coefficients. This work shows how asymptotic analysis can be used to develop and analyze models and numerical methods that are robust and work well for a wide range of parameters.


Reviews

“This book is a tutorial on numerical methods for multiscale problems accompanied by the corresponding numerical analysis. It is well studied as introductory material for graduate students at the master's level. ... All proofs are given in detail and results of instructive numerical experiments are demonstrated.” (Karsten Urban, Mathematical Reviews, May, 2018)



“This monograph is concerned with the asymptotic analysis and numerical methods for multiscale problems. … the book is orientated toward advanced undergraduate and graduate students.” (Rolf Dieter Grigorieff, zbMATH, Vol. 1366.65108, 2017)

Authors and Affiliations

  • Laboratório Nacional de Computação Científica-LNCC, Petrópolis, Brazil

    Alexandre L. Madureira

About the author

Alexandre L. Madureira is a senior researcher at the National Laboratory for Scientific Computing (LNCC), Brazil, and also works at the Brazilian School of Economics and Finance (EPGE) at the Fundação Getúlio Vargas (FGV). He holds a PhD from Penn State University, USA, and served as a visiting professor at the Istituto di Analisi Numerica (IAN), Italy, University of Colorado at Denver, USA, and Brown University, USA. His main field of interest is numerical analysis, more specifically modeling and analysis of multiscale problems from PDE and numerical points of view.

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