Numerical Models for Differential Problems

1. Front Matter

   Pages I-XVI

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2. A brief survey on partial differential equations

   Pages 1-10

3. Elements of functional analysis

   Pages 11-29

4. Elliptic equations

   Pages 31-59

5. The Galerkin finite element method for elliptic problems

   Pages 61-118

6. Parabolic equations

   Pages 119-138

7. Generation of 1D and 2D grids

   Pages 139-157

8. Algorithms for the solution of linear systems

   Pages 159-172

9. Elements of finite element programming

   Pages 173-216

10. The finite volume method

    Pages 217-226

11. Spectral methods

    Pages 227-269

12. Diffusion-transport-reaction equations

    Pages 271-312

13. Finite differences for hyperbolic equations

    Pages 313-344

14. Finite elements and spectral methods for hyperbolic equations

    Pages 345-382

15. Nonlinear hyperbolic problems

    Pages 383-400

16. Navier-Stokes equations

    Pages 401-455

17. Optimal control of partial differential equations

    Pages 457-500

18. Domain decomposition methods

    Pages 501-546

19. Reduced basis approximation for parametrized partial differential equations

    Pages 547-579

20. Back Matter

    Pages 581-601

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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. 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.

• Analysis
• Numerical modelling
• PDE
• algorithms
• finite element method
• hyperbolic equation
• partial differential equation

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)

• MOX, Department of Mathematics “F. Brioschi”, Politecnico di Milano, Milan, Italy

  Alfio Quarteroni

• CMCS-IACS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

  Alfio Quarteroni

The Author is Professor and Director of the Chair of Modelling and Scientific Computing (CMCS) at the Institute of Analysis and Scientific Computing of EPFL, Lausanne (Switzerland), since 1998, Professor of Numerical Analysis at the Politecnico di Milano (Italy) since 1989, and Scientific Director of MOX, since 2002. Author of 22 books published with Springer, and of about 200 papers published in refereed international Journals, Conference Proceedings and Magazines, Alfio Quarteroni is actually one of the strongest and reliable mathematicians in the world in the field of Modelling and SC.

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