Second-order error analysis on graded meshes for fractional Laplacian via Riesz fractional derivative

The high-order numerical analysis for 1D fractional Laplacian via the Riesz fractional derivative, under the low regularity solution, has presented significant challenges in the past decades. To fill in this gap, we analyse the local truncation errors by difference-quadrature method on graded meshes, which are far less than second-order convergence at the boundary layer. To prove the second-order global errors, we construct a suitable discrete barrier function and apply discrete maximum principle to the resulting matrix algebraic equation. We prove that the proposed scheme achieves second-order convergence on graded meshes even if the source term is singular or hypersingular. Numerical experiments illustrate the theoretical results.