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The method of Helsing and co-workers evaluates Laplace and related layerpotentials generated by a panel (composite) quadrature on a curve, efficientlyand with high-order accuracy for arbitrarily close targets. Since it exploitscomplex analysis, its use has been restricted to two dimensions (2D). We firstexplain its loss of accuracy as panels become curved, using a classical complexapproximation result of Walsh that can be interpreted as "electrostaticshielding" of a Schwarz singularity. We then introduce a variant that swaps thetarget singularity for one at its complexified parameter preimage; in thelatter space the panel is flat, hence the convergence rate can be much higher.The preimage is found robustly by Newton iteration. This idea also enables, forthe first time, a near-singular quadrature for potentials generated by smoothcurves in 3D, building on recurrences of Tornberg-Gustavsson. We apply this toaccurate evaluation of the Stokes flow near to a curved filament in the slenderbody approximation. Our 3D method is several times more efficient (both interms of kernel evaluations, and in speed in a C implementation) than the onlyexisting alternative, namely, adaptive integration.

Accurate quadrature of nearly singular line integrals in two and three dimensions by singularity swapping