  A new highly efficient method is developed for computation of traveling
periodic waves (Stokes waves) on the free surface of deep water. A convergence
of numerical approximation is determined by the complex singularites above the
free surface for the analytical continuation of the travelling wave into the
complex plane. An auxiliary conformal mapping is introduced which moves
singularities away from the free surface thus dramatically speeding up
numerical convergence by adapting the numerical grid for resolving
singularities while being consistent with the fluid dynamics. The efficiency of
that conformal mapping is demonstrated for Stokes wave approaching the limiting
Stokes wave (the wave of the greatest height) which significantly expands the
family of numerically accessible solutions. It allows to provide a detailed
study of the oscillatory approach of these solutions to the limiting wave.
Generalizations of the conformal mapping to resolve multiple singularities are
also introduced.
