Titlebook: Numerical Methods and Analysis of Multiscale Problems; Alexandre L. Madureira Book 2017 The Author(s) 2017 Asymptotic Analysis.Elliptic Eq

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Alexandre L. MadureiraOffers 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

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Introductory Material and Finite Element Methods,In this chapter, we introduce some notation, and also state some basic results regarding the Galerkin Method. In particular, some elementary estimates are presented, highlighting the importance of coercivity constants. This chapter also contains a brief introduction to some alternative methods, such as the ., the ., the ., the ., and . and ..

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Numerical Methods and Analysis of Multiscale Problems978-3-319-50866-5Series ISSN 2191-8198 Series E-ISSN 2191-8201

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One-Dimensional Singular Perturbed Problems,ensional advective dominated advection-diffusion problem, both in terms of numerical solutions and its asymptotic expansion. We then consider a more general asymptotic expansion, including a reaction term in the equation and considering the situation when the coefficients might depend on . as well.

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Two-Dimensional Reaction-Diffusion Equations,this time we show how to deal with the boundary layer in a two-dimensional problem, assuming that the boundary is smooth. We then derive an estimate for non-smooth domains. Finally, we present a numerical scheme that is a variation of the Residual Free Bubble method that works well for the problem under consideration.

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Partial Differential Equations with Oscillatory Coefficients,le one-dimensional case that still keeps most of the difficulties present in more sophisticated problems. We discuss three different approximation techniques: classical finite elements, homogenization, and Multiscale Finite Element methods (MsFEM). We show the advantages and pitfalls of each of the techniques, and present numerical results.

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