Finite difference method for inhomogeneous fractional Dirichlet problem

Abstract

We make the split of the integral fractional Laplacian as (−∆) s u = (−∆)(−∆) s−1 u, where s ∈ (0, 1 2 ) ∪ ( 1 2 , 1). Based on this splitting, we respectively discretize the one-and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an O(h 1+α−2s ) convergence rate is obtained when the solution u ∈ C 1,α ( Ωδ n ), where n is the dimension of the space, α ∈ (max(0, 2s − 1), 1], δ is a fixed positive constant, and h denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.

Error Estimates of Finite Difference Methods for the Fractional Poisson Equation with Extended Nonhomogeneous Boundary Conditions

“…Wu et al [23] proposed an efficient operator factorization method, where far field boundary conditions are approximated by numerical quadratures. Sun et al [19] considered a finite difference method for nonhomogeneous fractional Dirichlet problem with compactly supported boundary conditions. Huang and Oberman [11] developed a finite difference-quadrature method with asymptotic approximations of extended Dirichlet boundary condition.…”