The Numerical Solution of Boundary Integral Equations

Much of the research on the numerical analysis of Fredholm type integral equations during the past ten years has centered on the solution of boundary integral equations (BIE). A great deal of this research has been on the numerical solution of BIE on simple closed boundary curves S for planar regions. When a BIE is de ned on a smooth curve S, there are many numerical methods for solving the equation. The numerical analysis of most such problems is now well-understood, for both BIE of the rst and second kind, with many people having contributed to the area. For the case with the BIE de ned on a curve S which is only piecewise smooth, new numerical methods have been developed during the past decade. Such methods for BIE of the second kind were developed in the mid to late 80s; and more recently, high order collocation methods have been given and analyzed for BIE of the rst kind. The numerical analysis of BIE on surfaces S in R has become more active during the past decade, and we review some of the important results. The convergence theory for Galerkin methods for BIE is well-understood in the case that S is a smooth surface, for BIE of both the rst and second kind. For BIE of the second kind on piecewise smooth surfaces, important analyses have been given more recently for both Galerkin and collocation methods. In contrast, almost nothing is understood about collocation methods for solving BIE of the rst kind, regardless of the smoothness of S. Numerical methods for BIE on surfaces S in R lead to computationally expensive procedures, and a great deal of the research for such BIE has looked at the e cient numerical evaluation of integrals, the use of iterative methods for solving the associated linear systems, and the use of \fast matrix-vector calculations" for use in iteration procedures.