The finite element method and its applications. Transl. from the Japanese. (English) Zbl 0668.65081

Mori, Masatake

New York: Macmillan Co. XIV, 188 p.; $ 34.95 (1986).

This is an English translation, by the author, of his original text in Japanese. The book concentrates on the mathematical basis of the finite element method, and although a number of example equations are used to illustrate the application of the method there is no discussion of the numerical techniques to compute the final approximations.
The first chapter introduces the finite element method by using Galerkin’s method with piecewise linear basis functions to approximate a two-point boundary value problem by a system of tridiagonal equations. This example is extended in the following chapter to show that the method is equivalent to minimizing an associated functional. An introduction to the \(H_ 1\) norm and admissive functions then leads on to the fundamental theorem of variation and Euler’s equation. The incorporation of inhomogeneous Dirichlet and natural boundary conditions is considered. Chapter 3 concludes the discussion of the two-point boundary value problem by showing that the approximation is the best attainable in the energy norm, and error bounds are obtained for piecewise linear functions.
The next three chapters extend the application of the finite element method to elliptic equations in two space dimensions using linear and quadratic approximations over a triangulated domain. The incorporation of boundary conditions is dealt with both by extending the trial functions and by the use of penalty methods. Transformations into a standard triangle, isoparametric transformations and transformations of triangular elements with a curved side are discussed. This section concludes with an error analysis.
Chapter 7 considers the approximation of a fourth-order equation, modeling the displacement of a thin clamped plate, using quadratic basic functions. Nonconforming elements are discussed along with the associated error analyses. Finally the possibility of solving a fourth-order equation as a pair of coupled second-order equations is briefly mentioned.
The second half of the book deals in a similar fashion with time- dependent problems. The simple heat-conduction equation is used to show how the finite element method reduces the partial differential equation into a system of ordinary differential equations. A variety of crude time discretizations of the resultant system are considered. Error analyses and stability requirements are also covered. Other topics include the extension of the heat-conduction equation to two space dimensions, the diffusion equation with a convective term in one and two space dimensions and the use of upwindig, free boundary problems (using Stefan’s problem as an example), the wave equation and the Navier-Stokes equation. The book concludes with a chapter on the dual variational principle.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
74S05	Finite element methods applied to problems in solid mechanics
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
34B05	Linear boundary value problems for ordinary differential equations
65L10	Numerical solution of boundary value problems involving ordinary differential equations
65-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
35J25	Boundary value problems for second-order elliptic equations
35J40	Boundary value problems for higher-order elliptic equations
35K05	Heat equation
65N15	Error bounds for boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs

Keywords:

textbook; finite element method; Galerkin’s method; Euler’s equation; energy norm; error bounds; penalty methods; thin clamped plate; Nonconforming elements; fourth-order equation; second-order equations; heat-conduction equation; time discretizations; stability; diffusion equation; free boundary problems; Stefan’s problem; wave equation; Navier-Stokes equation