High-order Corrected Trapezoidal Rules for Functions with Fractional Singularities

10/08/2021

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by   Senbao Jiang, et al.

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In this paper, we introduce and analyze a high-order quadrature rule for evaluating the two-dimensional singular integrals of the forms I = ∫_R^2ϕ(x)x_1^2/|x|^2+α dx, 0< α < 2 where ϕ∈ C_c^N for N≥ 2. This type of singular integrals and its quadrature rule appear in the numerical discretization of Fractional Laplacian in the non-local Fokker-Planck Equations in 2D by Ha <cit.>. The quadrature rule is adapted from <cit.>, they are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is 2p+4-α, where p∈ℕ_0 is associated with total number of correction weights. Although we work in 2D setting, we mainly formulate definitions and theorems in n∈ℕ dimensions for the sake of clarity and generality.