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Numerical Analysis for Integral and Related Operator Equations: OT 52 (Operator Theory: Advances and Applications): Siegfried Proessdorf and Bernd Silbermann

About 15 years ago, when we were writing our booklet "Projection Methods and the Approximate Solution of Singular Equations" (1977) (that was the time when German native speakers still wrote in German, and so this little book is also in German), we were a long way from expecting that the numerical analysis of several classes of integral equations was on the point of entering a rapid stage of development, continuing until the present time and being certain to endure for a long time to come. This line of development was given decisive impetus by the increasing use of boundary element methods (BEM's, sometimes also referred to as boundary integral equation methods) in engineering and the natural sciences.

The essence of the methods alluded to may be described as follows. The solution to the boundary value problem under consideration is sought as an appropriately chosen integral over the boundary containing one or more unknown functions as well as a known fundamental solution of the differential equation to be solved. Inserting this ansatz (well known from potential theory for more than 100 years) into the boundary conditions leads to a boundary integral equation for the unknown function. The solution of the latter integral equation then provides the solution of the original boundary value problem in the form of an integral representation. The wide number of possible choices of boundary elements (e.g. of finite elements on the boundary as trial functions) and of discretizations of the boundary integral equations (e.g. collocation, Galerkin or quadrature methods) result in a whole variety of BEM's. Comparing BEM's with finite difference and finite element methods and considering the pros and cons, one will observe that BEM's take a series of advantages from the facts that they allow a reduction in the dimension of the problem by one unit and that they work equally well for both interior and exterior boundary value problems. Therefore, during the last decade BEM's have become a rather powerful and popular technique in engineering computations of boundary value problems arising from different fields of application. Notwithstanding the more than one hundred year usage of boundary integral equation methods in the analytical theory of boundary value problems, going back to C. Neumann's pioneering work in 1877, the mathematically rigorous foundation and error analysis of BEM's has been started on and (at least for two-dimensional problems) has made fairly satisfactory progress recently. The reason for this delay lies, in the author's view, in the fact that boundary integral operators are in general neither integral operators of the form identity plus compact operator nor of the form identity plus an operator with small norm, so that the existing standard theories for the numerical analysis of second kind Fredholm integral equations cannot be applied. Boundary reduction rather leads to singular integral equations, convolution equations (of Wiener-Hopf or Mellin type), or even to pseudodifferential equations. For instance, solving the Dirichlet problem for the Laplace equation in a domain with corners by a double layer potential ansatz amounts to a convolution equation of Mellin type of the form 5.0A) with the kernel 5.0C).

The study of the equations we encounter when applying BEM's requires having recourse to a series of heavy guns from mathematical analysis. So it is not surprising that this peculiarity is shared by the numerical analysis of these equations. In our opinion, the profound investigation of a broad variety of approximation methods for solving such integral equations is a presentday problem of numerical analysis. Due to the breadth and complexity of the questions touched upon above, it is impossible to coverall aspects of the matter by a single monograph. We therefore restrict our attention to the illumination of a few up-to-date methods and ideas, which, as we reckon, form an indispensable part of the numerical analysis of operator equations at present as well as in future. On the other hand, we are fully aware that the present book is nothing but a snapshot of what is going on and that its tone is set, moreover, by our own scientific interests.

The book is addressed to a wide audience of readers. We hope that both the mathematician interested in theoretical aspects of numerical analysis and the engineer wishing to see practically realizable recipes for computations will find a few suggestions. We tried to present the material in a form which allows any reader to go to the chapters or parts of the book he is interested in as quickly as possible. The interdependence table provides an overall view of the connection between the parts of this monograph.

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