  We study the problem of computing the exterior modulus of a bounded
quadrilateral. We reduce this problem to the numerical solution of the
Dirichlet-Neumann problem for the Laplace equation. Several experimental
results, with error estimates, are reported. Our main method makes use of an
$hp$-FEM algorithm, which enables computations in the case of complicated
geometry. For simple geometries, good agreement with computational results
based on the SC Toolbox, is observed. We also use the reciprocal error
estimation method introduced in our earlier paper to validate our numerical
results. In particular, exponential convergence, in accordance with the theory
of Babu\vska and Guo, is demonstrated.
