High-order corrected trapezoidal quadrature rules for functions with a logarithmic singularity in 2-D

In this report, we construct correction coefficients to obtain high-order trapezoidal quadrature rules to evaluate two-dimensional integrals with a logarithmic singularity of the form J(v) = ∫_D v(x, y) ln (√x² + y²) dx dy, where the domain D is a square containing the point of singularity (0,0) and v is a C^∞ function defined on the whole plane ℝ². The procedure we use is a generalization to 2-D of the method of central corrections for logarithmic singularities described in [1]. As in 1-D, the correction coefficients are independent of the number of sampling points used to discretize the square D. When v has compact support contained in D, the approximation is the trapezoidal rule plus a local weighted sum of the values of v around the point of singularity. These quadrature rules give an efficient, stable, and accurate way of approximating J(v). We provide the correction coefficients to obtain corrected trapezoidal quadrature rules up to order 20.