A method is proposed for evaluation of single and double layer potentials of the Laplace and Helmholtz equations on piecewise smooth manifold boundary elements with constant densities. The method is based on a novel two-term decomposition of the layer potentials, derived by means of differential geometry. The first term is an integral of a differential 2-form which can be reduced to contour integrals using Stokes' theorem, while the second term is related to the element curvature. This decomposition reduces the degree of singularity and the curvature term can be further regularized by a polar coordinate transform. The method can handle singular and nearly singular integrals. Numerical results validating the accuracy of the method are presented for all combinations of single and double layer potentials, for the Laplace and Helmholtz kernels, and for singular and nearly singular integrals.

翻译：针对分段光滑流形边界元上具有常密度的拉普拉斯与亥姆霍兹方程单层势和双层势，提出了一种求积方法。该方法基于微分几何推导的层势新型两项分解。第一项为可化为斯托克斯定理轮廓积分的微分2-形式积分，第二项与单元曲率相关。该分解降低了奇异性阶数，而曲率项可通过极坐标变换进一步正则化。方法可处理奇异积分和近奇异积分。针对拉普拉斯与亥姆霍兹核的单层势和双层势组合，以及奇异和近奇异积分，给出了验证方法精度的数值结果。