Fast Q1 finite element for two-dimensional integral fractional Laplacian

Abstract

To investigate 2-dimensional integral fractional Laplacian, a fast Q1 finite element method is proposed based on a weighted trapezoidal rule. Different from the very limited existing finite element approximations, the singular integrals are handled numerically, but the rest of integrals can be exactly evaluated. In addition, the entries of stiffness matrix with Toeplitz structure can be efficiently expressed as c2,sh2−γ144w·e with known vectors w and e, where γ∈(2s,2] and 0<s<1. We also prove the upper and lower bounds of the maximum and minimum eigenvalues of stiffness matrix and its condition number, and then study the error analysis of the developed scheme in discrete energy norm. Finally, a preconditioned conjugate gradient method coupled with fast Fourier transform (FFT)-based fast algorithm for matrix-vector product is provided to solve linear systems, and then some numerical experiments are carried out to verify the effectiveness of our scheme.