  We propose a novel mesh refinement scheme based on signal processing for
boundary integral simulations of inviscid droplet dynamics with axial symmetry.
A key idea is to directly access the Fourier coefficients of a principal
curvature as a function of the arclength through a natural change of variables.
The trapezoidal rule is applied to those Fourier-type integrals and the
resulting formula fits in the framework of the non-uniform fast Fourier
transform. This observation enables to efficiently use an envelope analysis and
smoothing filter to generate guidelines for mesh refinement in two singularity
formation scenarios. Applications also include a non-iterative construction of
the uniform parametrization for an important class of plane curves, which is
used in a convergence study of the time-stepping procedure implemented in the
previous work by Nitsche and Steen [J. Comput. Phys. 200 (2004) 299].
