Fast direct solvers for integral equations in complex three-dimensional domains

This work was supported in part by the Applied Mathematical Sciences Program of the US Department of Energy under Contract DEFG0200ER25053. † This work was supported by DARPA through the Protein Design Processes Program (DSO contract HR0011-05-1-0044). Present address: NASA GISS and Columbia University, 2880 Broadway, New York, NY 10025. ‡ This work was supported in part by National Science Foundation grants DMS-0748488 and DMS-0610097. § This work was supported in part by AFOSR grant FA9550-07-1-0541. Methods for the solution of boundary integral equations have changed significantly during the last two decades. This is due, in part, to improvements in computer hardware, but more importantly, to the development of fast algorithms which scale linearly or nearly linearly with the number of degrees of freedom required. These methods are typically iterative, based on coupling fast matrix-vector multiplication routines with conjugate-gradient-type schemes. Here, we discuss methods that are currently under development for the fast, direct solution of boundary integral equations in three dimensions. After reviewing the mathematical foundations of such schemes, we illustrate their performance with some numerical examples, and discuss the potential impact of the overall approach in a variety of settings