# Integral equations

Despite decades of zealous effort, there are remarkably few

applications where fast integral equation solvers dominate. To help

explain why, we will describe effective strategies, assess performance,

and present remaining challenges associated with applying fast solvers

to problems in interconnect model extraction, biomolecular electrostatics,

bodies in microflows, nanophotonics, and Casimir force calculation.

iterative solver. I show how to alleviate them in certain situations.

Then I turn to the main topic of the talk – a method to enhance the

efficiency of integral equation based schemes for elliptic PDEs on

domains with corners, multi-wedge points, and mixed boundary

conditions. The key ingredients are a block-diagonal inverse

preconditioner 'R' and a fast recursion, 'i=1,...,n', where step 'i'

inverts and compresses contributions to 'R' from the outermost

for the fast and accurate solution of boundary integral

equations on two-dimensional domains whose boundaries

have corner points. Our approach has two key advantages

over existing and recently suggested schemes: (1) it

does not require a prior analytic estimates for solutions,

and (2) many aspects of the scheme generalize readily

to singular three-dimensional domains.