Numerical differential equations. Theory and technique, ODE methods, finite differences, finite elements and collocation (Q5890548)

The book is subdivided in two main parts: Part 1. Modeling and visualization (6 chapters, 186 pages), Part 2. Methods and theory (4 chapters, 163 pages). The first part focusses on applications and techniques, while in the second part the theory is presented.NEWLINENEWLINEThe author puts much emphasis on presenting interesting examples in modeling and simulating processes coming from different fields of applications as mechanical and civil engineering, biology, finances, and at the same time developing in a first step the basic ideas for applying numerical methods for their practical solution, many of them with the results illustrated in figures. Especially, there is to mention the flow about an airfoil, population models (predator/prey and herding instinct with random parameters), cell chemotaxis, Stokes equation, river pollution, traffic congestion, herd formation and 18 pages dedicated to the Black-Scholes equation for options pricing.NEWLINENEWLINEComing to the second part, the reader will immediately note quite a number of misprints. One may suspect that not so much care was paid when writing down the material. And, unfortunately, also the mathematics suffer severely from a lack of accuracy. A reason may be the author's intention as stated in the foreword ``to provide the reader or student with sufficient background to be able to access the numerical analysis literature and thereby develop the deep understanding of those settings of particular interest'' is too ambitious. Lax-Richtmyer, finite element spaces, Sobolev spaces including imbedding theorems and boundary traces, piecewise polynomial interpolation and its approximation power, Bramble-Hilbert, Lax-Milgram, Céa's lemma, convergence of the finite element method (FEM) and collocation, frequently including proofs, is too much material for 163 pages. This review is not a proof-reading, but I will give one example standing representative for many others. Céa's lemma is, as usual, used to prove convergence of the FEM for a second-order elliptic equation subject to Dirichlet boundary conditions. But not the energy norm is taken for obtaining the standard errror estimates but the \(L^2\) norm is used instead. Also, only the case of the exact solution lying in \(H^k\) for a sufficiently large \(k\) (guaranteeing its continuity) is considered.