Variational Methods for Boundary Integral Equations: Theory and Applications

Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. The later has become one of the most popular numerical schemes in recent years. In this expository paper, we discuss some of the essential features of the methods, their intimate relations with the variational formulations of the corresponding partial differential equations and recent developments with respect to applications in domain composition from both mathematical and numerical points of view. 1 Variational Formulations for Boundary Integral Equations It is well known that the reduction of boundary value problems to integral equations is by no means a unique process. In spite of many formulations, there are two kinds of boundary integral equations, the first and the second kind boundary integral equations either from the direct or indirect approach. The variational formulations of boundary integral equations in general may depend upon the types of boundary integral equations under consideration. In the following we shall systematically discuss these formulations through simple model problems and present some of the basic theorems and indicate typical ways for obtaining these results which may be employed in general for the variational methods concerning boundary integral equations. ∗Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA; Email: [email protected]. To appear in the Proceedings of the Taormina Symposium in memory of Professor G. Fichera.