  We study the problem of computing the conformal modulus of rings and
quadrilaterals with strong singularities and cusps on their boundary. We reduce
this problem to the numerical solution of the associated Dirichlet and
Dirichlet-Neumann type boundary values problems for the Laplace equation.
Several experimental results, with error estimates, are reported. In
particular, we consider domains with dendrite like boundaries, in such cases
where an analytic formula for the conformal modulus can be derived. Our
numerical method makes use of an $hp$-FEM algorithm, written for this very
complicated geometry with strong singularities.
