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Numerical Models for Differential Problems Hardcover – Apr 8 2009

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From the reviews: “This book contains the basic concepts for the approximation of differential equations which arise in the mathematical modeling of real life applications. … it can be used as a textbook for graduate-level courses. Moreover, the interested reader can find a lot of information on the various aspects of the numerical approximation of differential problems, so that it can also be used as a starting point for the study of more specific topics in this field.” (Lucia Gastaldi, Mathematical Reviews, Issue 2010 h)

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In this text, we introduce the basic concepts for the numerical modelling of partial  differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. We then analyze numerical solution methods based on finite elements, finite differences, finite volumes, spectral methods and domain decomposition methods, and reduced basis methods. 

In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs.

The text does not require any previous advanced mathematical knowledge of partial differential equations: the absolutely essential concepts are reported in a preliminary chapter. It is therefore suitable for students of bachelor and master courses in scientific disciplines, and recommendable to those researchers in the academic and extra-academic domain who want to approach this interesting branch of applied mathematics.

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An important theoretical reference to Galekin methods for PDEs Oct. 20 2010
By Simone Marras - Published on
Format: Hardcover
Written by one of the leading figures in the field of variational methods and numerical mathematics, this book is an invaluable reference for graduate students and researchers working in the field of Galerkin methods. It is written by a mathematician, with the formalism proper of a mathematician; it may thus appear somewhat cumbersome if the reader is seeking an applied insight of the finite element method. However, the inclined engineer/scientist/mathematician will find its formalism an important ingredient to deeply understand the theory behind the method, although built in a very understandable language.
The books covers different types of Galerkin schemes, with special attention to finite elements and spectral elements, and a bit on the finite volume and finite difference methods. It should be considered by those who are willing to make an intellectual effort to understanding the theory of finite elements, but should be avoided by those expecting a book that could help with their practical implementation --some implementation issues are introduced in the context of object-oriented programming, but it may not be enough for beginners. This is the translation of the original Italian edition: Modellistica Numerica per Problemi Differenziali.