Arbitrarily high-order trapezoidal rules for functions with fractional singularities in two dimensions

摘要

In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the formsIi,j=∫R2ϕ(x)xixj|x|2+αdx,0<α<2where i,j∈{1,2} and ϕ∈CcN for N≥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−α, where p∈N0 is associated with total number of correction weights. Although we work in 2D setting, we formulate definitions and theorems in n∈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.