High-order Corrected Trapezoidal Rules for Functions with Fractional Singularities

In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms \begin{align} I_{i,j} = \int_{\mathbb{R}^2}\phi(x)\frac{x_ix_j}{|x|^{2+\alpha}} \d x, \quad 0< \alpha < 2 \end{align} where \(i,j\in\{1,2\}\) and \(\phi\in C_c^N\) for \(N\geq 2\). This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is \(2p+4-\alpha\), where \(p\in\mathbb{N}_{0}\) is associated with total number of correction weights. Although we work in 2D setting, we formulate definitions and theorems in \(n\in\mathbb{N}\) dimensions when appropriate for the sake of generality. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules.