Numerical Methods and Analysis of Multiscale Problems

Description

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. Front Matter....Pages i-x Introductory Material and Finite Element Methods....Pages 1-21 One-Dimensional Singular Perturbed Problems....Pages 23-38 An Application in Neuroscience: Heterogeneous Cable Equation....Pages 39-47 Two-Dimensional Reaction-Diffusion Equations....Pages 49-65 Modeling PDEs in Domains with Rough Boundaries....Pages 67-84 Partial Differential Equations with Oscillatory Coefficients....Pages 85-107 Back Matter....Pages 109-123