首页|Arbitrarily high-order trapezoidal rules for functions with fractional singularities in two dimensions
Arbitrarily high-order trapezoidal rules for functions with fractional singularities in two dimensions
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NSTL
Elsevier
In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms I-i,I-j = integral(R2) phi(x) x(i)x(j)/vertical bar x vertical bar(2+alpha) dx, 0 < alpha < 2 where i, j is an element of {1, 2} and phi is an element of C-c(N) 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 - alpha, where p is an element of N-0 is associated with total number of correction weights. Although we work in 2D setting, we formulate definitions and theorems in n is an element of 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. (C) 2022 Elsevier Inc. All rights reserved.
NSTL
Elsevier
