  We apply domain functionals to study the conformal capacity of condensers
$(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is
a compact subset of $G$. Due to conformal invariance, our main tools are the
hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$.
Novel computational algorithms based on implementations of the fast multipole
method are combined with analytic techniques. Computational experiments are
used throughout to, for instance, demonstrate sharpness of established
inequalities. In the case of model problems with known analytic solutions, very
high precision of computation is observed.
