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Mathematical and numerical methods for partial differential equations. Applications to engineering sciences. (Méthodes mathématiques et numériques pour les équations aux dérivées partielles. Applications aux sciences de l’ingénieur.) (French)
Paris: Lavoisier (ISBN 978-2-7430-1480-3/hbk). vii, 376~p. (2013).
The book under review contains seven chapters. Chapter 1, Introduction in the methods of functional analysis with applications to PDE, contains six paragraphs, which presents the fundamental problems of functional analysis used in solving of the partial differential equations, using the finite element method: 1.1 Fundamental problems of mathematical analysis, 1.2 Banach spaces, 1.3 Hilbert Spaces, 1.4 Elementary knowledge of distributions theory, 1.5 Sobolev spaces, 1.6 Fundamental theorems of functional analysis for PDE. In Chapter 2, The finite element method, the finite element method is presented in order to apply it to approximate the solution of partial differential equations. So, the variational form of the Poisson equation with Dirichlet boundary conditions is presented, the existence, uniqueness and regularity of the solution of these variational equation, the equivalence between strong formulation and weak formulation (variational problem). Chapter 2 includes also variational formulations, the convergence of the finite element method, description of the usual finite element (finite elements in one-dimensional space, finite elements in two-dimensional space and in three-dimensional space). Chapter 3 is named Variational formulation of elliptic boundary problems. Here three basic problems, described by PDEs, are presented. These are: 3.1 Thermal problem with mixed boundary conditions, which models the heat exchange between a fluid occupying a bounded domain and the external environment. It shows the variational form of the problem, the existence and uniqueness of the solution of variational equation and a priori estimation of the solution. 3.2 Thermal conductivity problems with Neumann boundary conditions. In the last paragraph of this chapter: 3.3 Stokes problem for incompressible environments is differs from the first two because besides boundary conditions, EDP has constraints and within the domain of definition of the solution ($\operatorname{div} u = 0$). These section shows the variational form of the problem, the existence, uniqueness and a priori estimation of the solution. Chapter 4, Finite elements and boundary value problems reference, describes four boundary value problems reference in one-dimensional case (for ordinary differential equations of order 2). For all four benchmark problems the following steps are considered: variational formulation (or weak formulation) for the continuous problem (or the strong form), the existence and uniqueness of weak solution using the Max-Milgram theorem, regularity of the weak solution, writing the system of nodal equations which provides the variational weak solution of the problem. The four benchmark problems are: 4.1 The Dirichlet problem, a problem with Dirichlet boundary conditions of a 2nd order ordinary differential equations. 4.2 The Neumann problem is a problem with Neumann boundary conditions of a 2nd order ordinary differential equations. 4.3 The Fourier-Dirichlet problem is a problem with mixed boundary conditions Fourier-Dirichlet of a 2nd order ordinary differential equations. 4.4 The periodic problem is a problem with periodic boundary conditions, of a 2nd order ordinary differential equations. Chapter 5, Finite elements in deformable solid mechanics, contains two paragraphs. 5.1 Mixed problem, lies in the finite element approximation of linear elastostatics problem with mixed boundary conditions. Is written variational formulation associated Navier equation, and is approximated with the Lagrange finite elements of degree one in 2D. In Section 5.2 Clamped plate, is followed the same steps as in the case of the paragraph 5.1, but for solving a 4th order PDE (biharmonic operator) with null boundary conditions for the solution and its derivative with respect to the outward normal. Chapter 6, Finite elements applied to materials resistance is devoted to finite element analysis of classic examples of strength of materials, namely: the rod recessed at one end subjected to tractions along, the rod recessed at one end and supported at the other end and subjected to a transverse force and the rod recessed at both ends (Euler-Bernoulli’s theory) subjected to transverse forces. Chapter 7, Finite elements applied to nonlinear problems approximates, with the finite element method, the solution of Burgers equation with viscosity, the non-linear integro-differential equations and the Riccati differential equation.
Reviewer: Nicolae Pop (Baia Mare)
Classification: N15 M55 I95
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