Efficient Cubature Rules for the Numerical Integration of Logarithmic Singularities

In this work, we aim to improve the efficiency of a higher-order boundary element method for solving axi-symmetric electrostatic problems. To do so, we generate a set of tailored two-dimensional cubature rules for the efficient numerical integration of logarithmic singularities. In a series of numerical experiments, we demonstrate that our cubature rules can reduce the matrix setup time by a factor of two when compared to a more conventional tensor-product approach. This allows a significant reduction in computational time for the reconstruction of dielectric properties in Near-Field Scanning Microwave Microscopy. Furthermore, our approach is by no means limited to axi-symmetric systems or logarithmic singularities. The procedure can be readily generalized to the four-dimensional integrals occurring in fully three-dimensional computations and it can also be modified to handle stronger singularities.