Integral equation methods for elastance and mobility problems in two dimensions

We present a new, unified approach to the solution of the elastance problem in electrostatics and the mobility problem in Stokes flow. More precisely, we construct integral representations that lead to resonance-free Fredholm integral equations of the second kind and well-conditioned linear systems upon discretization. By coupling our integral equations with high order quadrature and fast multipole acceleration, large-scale problems can be solved with only modest computing resources. We also discuss some applications of these boundary value problems in applied physics.