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We present a simple and effective method for evaluating double-andsingle-layer potentials for Laplace's equation in three dimensions close to theboundary. The close evaluation of these layer potentials is challenging becausethey are nearly singular integrals. The method we propose is based on writingthese layer potentials in spherical coordinates where the point at which theirkernels are peaked maps to the north pole. An N-point Gauss-Legendre quadraturerule is used for integration with respect to the the polar angle rather thanthe cosine of the polar angle. A 2N-point periodic trapezoid rule is used tocompute the integral with respect to the azimuthal angle which acts as anatural and effective averaging operation in this coordinate system. Thenumerical method resulting from combining these two quadrature rules in thisrotated coordinate system yields results that are consistent with asymptoticbehaviors of the double- and single-layer potentials at close evaluationdistances. In particular, we show that the error in computing the double-layerpotential, after applying a subtraction method, is quadratic with respect tothe evaluation distance from the boundary, and the error is linear for thesingle-layer potential. We improve upon the single-layer potential byintroducing an alternate approximation based on a perturbation expansion andobtain an error that is quadratic with respect to the evaluation distance fromthe boundary.

Close evaluation of layer potentials in three dimensions