In this paper, we introduce and analyze a high-order quadrature rule for evaluating the two-dimensional singular integrals of the forms \[ I = \int_{R^2}\phi(x)\frac{x_1^2}{|x|^{2+\alpha}} dx, \quad 0< \alpha < 2 \] where $\phi\in C_c^N$ for $N\geq 2$. This type of singular integrals and its quadrature rule appear in the numerical discretization of Fractional Laplacian in the non-local Fokker-Planck Equations in 2D by Ha \cite{HansenHa}. The quadrature rule is adapted from \cite{MarinTornberg2014}, they 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 mainly formulate definitions and theorems in $n\in\mathbb{N}$ dimensions for the sake of clarity and generality.
翻译:在本文中,我们引入并分析一个高阶二次曲线规则,用于评价表格的二维单元集成[I =\ int ⁇ R ⁇ 2 ⁇ 2 ⁇ phi(x)\frac{x_1 ⁇ 2 ⁇ x ⁇ 2 ⁇ 2 ⁇ alpha ⁇ dx,\qad 0 <\alpha < 2\]\alpha < 2\\\\\\\\\\\\\\\\>,其中美元=C_c ⁇ C_N$=2美元。这种单元集成及其二次曲线规则出现在非本地Fokker-Planck方程式中Freactional Laplacian数字分解的数值中,由Ha\cite{HansenHa}在 2D 中显示。根据\ cite{ MarinTornberg2014} 调整的二次曲线规则,它们具有对奇点的校正权重值。 我们证明趋近2p+4\\\\ alphain$, 其中的值与整重总重数有关。