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A new highly efficient method is developed for computation of travelingperiodic waves (Stokes waves) on the free surface of deep water. A convergenceof numerical approximation is determined by the complex singularites above thefree surface for the analytical continuation of the travelling wave into thecomplex plane. An auxiliary conformal mapping is introduced which movessingularities away from the free surface thus dramatically speeding upnumerical convergence by adapting the numerical grid for resolvingsingularities while being consistent with the fluid dynamics. The efficiency ofthat conformal mapping is demonstrated for Stokes wave approaching the limitingStokes wave (the wave of the greatest height) which significantly expands thefamily of numerically accessible solutions. It allows to provide a detailedstudy of the oscillatory approach of these solutions to the limiting wave.Generalizations of the conformal mapping to resolve multiple singularities arealso introduced.

New conformal mapping for adaptive resolving of the complex singularities of Stokes wave