A Fast Method for the Off-Boundary Evaluation of Laplace Layer Potentials by Convolution Sums

Abstract

In off-boundary computations of layer potentials, the near-singularities in integrals near the boundary presents challenges for conventional quadrature methods in achieving high precision. Additionally, the significant complexity of O(n2) interactions between n targets and n sources reduces the efficiency of these methods. A fast and accurate numerical algorithm is presented for computing the Laplace layer potentials on a circle with a boundary described by a polar curve. This method can maintain high precision even when evaluating targets located at a close distance from the boundary. The radial symmetry of the integral kernels simplifies their description. By exploiting the polar form of the boundary and applying a one-dimensional exponential sum approximation along the radial direction, an approximation of layer potentials by the convolution sum is obtained. The algorithm uses FFT convolution to accelerate computation and employs a local quadrature to maintain accuracy for nearly singular terms. Consequently, it achieves spectral accuracy in regions outside of a sufficiently small neighborhood of the boundary and requires O(nlogn) arithmetic operations. With the help of this algorithm, layer potentials can be efficiently evaluated on a computational domain.

Keywords

• layer potential
• nearly singular integrals
• numerical integration
• approximation by exponential sums