Solving Numerical Pdes: Problems, Applications, Exercises

✍ Scribed by Numerisches Verfahren; Veneziani, Alessandro; Saleri, Fausto; Formaggia, Luca

• Publisher: Springer
• Year: 2011;2012
• Tongue: English
• Leaves: 439
• Category: Library

✦ Synopsis

This book stems from the long standing teaching experience of the authors in the courses on Numerical Methods in Engineering and Numerical Methods for Partial Differential Equations given to undergraduate and graduate students of Politecnico di Milano (Italy), EPFL Lausanne (Switzerland), University of Bergamo (Italy) and Emory University (Atlanta, USA). It aims at introducing students to the numerical approximation of Partial Differential Equations (PDEs). One of the difficulties of this subject is to identify the right trade-off between theoretical concepts and their actual use in practice. With this collection of examples and exercises we try to address this issue by illustrating "academic" examples which focus on basic concepts of Numerical Analysis as well as problems derived from practical application which the student is encouraged to formalize in terms of PDEs, analyze and solve. The latter examples are derived from the experience of the authors in research project developed in collaboration with scientists of different fields (biology, medicine, etc.) and industry. We wanted this book to be useful both to readers more interested in the theoretical aspects and those more concerned with the numerical implementation.

✦ Table of Contents

Title Page......Page 0
Copyright Page......Page 4
Preface......Page 5
Table of Contents......Page 8
Part I Basic Material......Page 10
1 Some fundamental tools......Page 11
1.1 Hilbert spaces......Page 12
1.2 Distributions......Page 13
1.3 The spaces Lp and Hs......Page 14
1.4 Sequences in lp......Page 18
1.5 Important inequalities......Page 19
1.6 Brief overview of matrix algebra......Page 21
2.1 The one dimensional case: approximation by piecewise polynomials......Page 24
2.2.1 Geometric preliminary definitions......Page 29
2.2.2 The finite element......Page 36
2.2.3 Parametric Finite Elements......Page 42
2.2.4 Function approximation by finite elements......Page 51
2.3.1 Difference quotients in dimension one......Page 53
Part II Stationary Problems......Page 70
3 Galerkin-finite element method for elliptic problems......Page 71
3.1 Approximation of 1D elliptic problems......Page 73
3.1.1 Finite differences in 1D......Page 95
3.2 Elliptic problems in 2D......Page 106
3.3.1 Overlapping methods......Page 137
3.3.2 Non-overlapping methods......Page 147
4 Advection-diffusion-reaction (ADR) problems......Page 153
4.1 Preliminary problems......Page 155
4.2 Advection dominated problems......Page 163
4.3 Reaction dominated problems......Page 196
Part III Time Dependent Problems......Page 209
5 Equations of parabolic type......Page 210
5.1 Finite difference time discretization......Page 211
5.2 Finite element time discretization......Page 268
6.1 Scalar advection-reaction problems......Page 281
6.2 Systems of linear hyperbolic equations of order one......Page 314
7 Navier-Stokes equations for incompressible fluids......Page 336
7.1 Steady problems......Page 337
7.2 Unsteady problems......Page 369
Part IV Appendices......Page 395
A.1 Storing techniques for sparse matrices......Page 396
A.1.1 The COO format......Page 400
A.1.2 The skyline format......Page 402
A.1.3 The CSR format......Page 405
A.1.4 The CSC format......Page 406
A.1.5 The MSR format......Page 407
A.2.1 Elimination of essential degrees of freedom......Page 410
A.2.2 Penalization technique......Page 411
A.2.3 “Diagonalization” technique......Page 412
A.2.4 Essential conditions in a vectorial problem......Page 414
B Who’s who......Page 417
References......Page 425
Subject Index......Page 430