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Numerical methods and analysis of multiscale problems. (English) Zbl 1366.65108

SpringerBriefs in Mathematics; SBMAC SpringerBriefs. Cham: Springer; Rio de Janeiro: Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC) (ISBN 978-3-319-50864-1/pbk; 978-3-319-50866-5/ebook). x, 123 p. (2017).
This monograph is concerned with the asymptotic analysis and numerical methods for multiscale problems. The book is subdivided into the following six chapters:
Chapter 1: Introductory material and finite element methods (21 pages)
Chapter 2: One-dimensional singularly perturbed problems (16 pages)
Chapter 3: An application to neuroscience: heterogeneous cable equation (9 pages)
Chapter 4: Two-dimensional reaction-diffussion equation (17 pages)
Chapter 5: Modeling PDEs in domains with rough boundaries (18 pages)
Chapter 6: Partial differential equations with oscillatory coefficients (23 pages)
completed by 201 references and an index.
In Chapter 1, several finite element methods are introduced. Among them is the multiscale finite element method, where the trial functions are local solutions of the original problem, which turns out to be successfully applied to the problems in the following chapters. Most of the chapters are organized by presenting an asymptotic analysis (also in the two-dimensional case of Chapter 4, Chapters 2, 3, 5 and 6 are set in one dimension) followed by some numerical methods including numerical results and corresponding figures illustrating their efficiency. In Chapter 3, a singularly perturbed second-order differential equation is considered with the peculiarity that the potential is a linear combination of Dirac’s delta function. Error estimates are given both for the truncated asymptotic expansions and the finite element approximations, mostly including proofs. The proofs in Chapter 4–6 are brief and frequently relying on mathematical tools not covered in Chapter 1 but equipped with references. As the author says in the Introduction, the book is orientated toward advanced undergraduate and graduate students.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Z05 Applications to the sciences
35B25 Singular perturbations in context of PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
92C20 Neural biology
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