Numerical solution of partial differential equations by the finite element method. (English) Zbl 0628.65098

Johnson, Claes

Cambridge etc.: Cambridge University Press. 278 p. (Orig. Studentlitteratur, Lund, Sweden) (1987).

“The purpose of this book is to give an easily accessible introduction to the finite element method as a general method for the numerical solution of partial differential equations in mechanics and physics covering all the three main types of equations namely elliptic, parabolic and hyperbolic equations.”
The author keeps the mathematical framework as simple as possible but generously illustrated by a great deal of explanatory examples. The work emphasizes the numerical aspects of the finite element method and many applications not only to linear problems but also to some nonlinear problems (compressible flow, incompressible Navier-Stokes equations, a nonlinear parabolic problem etc.) are considered.
Besides classical aspects of the finite element method, related to elliptic equations, the book treats some parabolic and hyperbolic problems using recent results of the author based on a discontinuous Galerkin and streamline diffusion type finite element method. In particular, finite elements for the time discretization are used as well. All proposed methods are accompanied by error estimates and very detailed discussions. The work also contains a chapter on the boundary element method connected to elliptic problems (finite element methods for Fredholm equations of the first and second kind are exposed). Two special chapters are devoted to a mixed finite element method to respectively, to curved elements and to the effect of numerical integration (quadrature formulas) on the accuracy of the finite element method.
The most important and the most original part of book seems to be that related to hyperbolic problems (time-dependent convection-diffusion problems, Friedrichs’ systems etc.). Actually the whole work opens new possibilities in proper understanding of the mathematical structure and features of the finite element method. It represents an important achievement in the literature on the finite element method.

Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65K05	Numerical mathematical programming methods
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25	Boundary value problems for second-order elliptic equations
65N15	Error bounds for boundary value problems involving PDEs
65F05	Direct numerical methods for linear systems and matrix inversion
65F10	Iterative numerical methods for linear systems
65N40	Method of lines for boundary value problems involving PDEs
35K05	Heat equation
35L20	Initial-boundary value problems for second-order hyperbolic equations
35C15	Integral representations of solutions to PDEs
65R20	Numerical methods for integral equations
90C25	Convex programming
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

convex programming; elementary textbook; finite element method; compressible flow; incompressible Navier-Stokes equations; error estimates; boundary element method; time-dependent convection-diffusion problems; Friedrichs’ systems