  We study numerical conformal mapping of multiply connected planar domains
with boundaries consisting of unions of finitely many circular arcs, so called
polycircular domains. We compute the conformal capacities of condensers defined
by polycircular domains. Experimental error estimates are provided for the
computed capacity and, when possible, the rate of convergence under refinement
of discretisation is analysed. The main ingredients of the computation are two
computational methods, on one hand the boundary integral equation method
combined with the fast multipole method and on the other hand the $hp$-FEM
method. The results obtained with these two methods agree with high accuracy.
