Numerical Methods and Analysis of Multiscale Problems

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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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.

• Asymptotic Analysis
• Elliptic Equations
• Finite Elements
• Multiscale
• Singular Perturbed Problems
• partial differential equations

“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)

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

  Alexandre L. Madureira

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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