Survey lectures on the mathematical foundations of the finite element method. With the collaboration of G. Fix and R. B. Kellogg.

*(English)*Zbl 0268.65052
Math. Found. finite Elem. Method Appl. part. differ. Equations, Sympos. Univ. Maryland, Baltimore 1972, 1-359 (1972).

This work arises as a result of a series of ten lectures included in the program of Symposium at the University of Maryland and is an attempt to focus on the theoretical foundations of the finite element method (another name is variation-difference method).

As the authors emphasized, the themes of lectures were various, the lectures did not exhaust the question under discussion and therefore the work could not be considered as a monograph on the finite element method (FEM). The manuscript consists of ten chapters.

The first chapter is introductory. The second and third chapters contain the theory of Sobolev spaces and a part of the theory of the elliptic boundary value problem (as a rule – without proofs). A central place is occupied by the forth chapter with theorems on approximations of Sobolev spaces by means of local functions (another name is Hill function).

In the fifth chapter the authors give a highly detailed account of different variational principles in the case of the boundary value problem for second order equations and also touch variational principles in the case of higher order equations and systems.

The sixth chapter is devoted to the construction of schemes of FEM by means of different variational principles. The penalty-method which attracts the mathematician’s attention during recent times and also the weighted least squares method are discussed in the seventh chapter. FEM for nonregular coefficients and boundary (for Lipschitzian boundary and piecewise smooth boundary) is considered in the eight chapter.

The ninth chapter deals with stability of the corresponding equations. The tenth chapter is devoted to eigenvalue problems; here results follow from theorems of approximation and works of Vainikko. Finally the eleventh chapter contains an original account of FE for time dependent problems.

All chapters have a bibliography, which however does not supply exhaustive notion about results that are in the literature and therefore cannot serve as a source for historical investigation.

For the entire collection see [Zbl 0259.00014].

As the authors emphasized, the themes of lectures were various, the lectures did not exhaust the question under discussion and therefore the work could not be considered as a monograph on the finite element method (FEM). The manuscript consists of ten chapters.

The first chapter is introductory. The second and third chapters contain the theory of Sobolev spaces and a part of the theory of the elliptic boundary value problem (as a rule – without proofs). A central place is occupied by the forth chapter with theorems on approximations of Sobolev spaces by means of local functions (another name is Hill function).

In the fifth chapter the authors give a highly detailed account of different variational principles in the case of the boundary value problem for second order equations and also touch variational principles in the case of higher order equations and systems.

The sixth chapter is devoted to the construction of schemes of FEM by means of different variational principles. The penalty-method which attracts the mathematician’s attention during recent times and also the weighted least squares method are discussed in the seventh chapter. FEM for nonregular coefficients and boundary (for Lipschitzian boundary and piecewise smooth boundary) is considered in the eight chapter.

The ninth chapter deals with stability of the corresponding equations. The tenth chapter is devoted to eigenvalue problems; here results follow from theorems of approximation and works of Vainikko. Finally the eleventh chapter contains an original account of FE for time dependent problems.

All chapters have a bibliography, which however does not supply exhaustive notion about results that are in the literature and therefore cannot serve as a source for historical investigation.

For the entire collection see [Zbl 0259.00014].

Reviewer: Ju. Demjanowitsch (Yu. K. Dem’yanovich)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |