## Boundary integral equation analyses of singular, potential, and biharmonic problems.(English)Zbl 0553.76001

Lecture Notes in Engineering, 7. Berlin etc.: Springer-Verlag. IV, 173 p. DM 32.00; \$ 11.70 (1984).
This work is concerned largely, but not exclusively, with the authors’ applications of the boundary element method to plane biharmonic problems arising in fluid flow at zero Reynold’s numbers. The solution of the biharmonic equation is obtained by solving numerically a pair of coupled integral equations for whichever of the stream function, vorticity and their fluxes are unknown on the boundary of the solution domain.
In chapters 2 and 3 the authors describe how the basic boundary element method may be modified, using a singularity subtraction technique, so as to enable boundary singularities in the unknown field variables to be treated numerically. The basic and modified methods are successfully applied to a number of specific problems; i.e., stick-slip flow between parallel plates, circulating flow induced in a rectangular cavity by a moving boundary and flow in a stepped channel.
In chapter 4 the free boundary problem of a jet issuing from between parallel plates is considered. This problem is intrinsically nonlinear and an efficient iterative method is presented for establishing the location of the free boundary. In chapter 6 the problem of flow in bearings with large gap geometries is successfully treated.
The contents of the above chapters convincingly demonstrate the practical efficiency and accuracy of the boundary element method for biharmonic problems.
In chapter 4 a nonlinear elliptic equation, subject to nonlinear boundary conditions and involving boundary singularities, is considered. This problem arises in steady state heat flow in materials with temperature dependent thermal conductivity. The nonlinear elliptic operator is reduced to the Laplacian by means of the Kirchhoff transformation. An alternative, and more efficient method for the numerical treatment of boundary singularities is given. This consists of using singular approximating basis functions over those elements adjacent to the singular point. An implementation of the Newton-Raphson iterative method is described for dealing with the nonlinear boundary conditions. Unfortunately this chapter contains a small but significant mistake in the definition of the Kirchhoff transformation (Eqn. (6) should read $$T=g(\phi))$$ and this mistake is repeate throughout all the relevant equations presented.
The whole work may be seen to be largely a collection of the authors’ papers, with the addition of introductory and concluding chapters. This inevitably involves a considerable amount of needless repetition, since in each chapter there is a tendency to repeat the governing equations and explanations of notation, terminology and motivating ideas. The work does not represent a survey of the state of the art with regard to the numerical treatment of singularities or the numerical solution of integral equations related to biharmonic problems. It is, however, a clear description of the particular methods used by the authors. The problems presented are interesting and would be difficult to solve accurately by other methods. The work is, therefore, to be welcomed as a useful contribution to the literature on applications of the boundary element method.
Reviewer: D.M.Hough

### MSC:

 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76M99 Basic methods in fluid mechanics 65Z05 Applications to the sciences 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs