Numerical analysis and efficient implementation of fast collocation methods for fractional Laplacian model on nonuniform grids

TL;DR

This paper introduces a fast collocation method using Krylov subspace solvers for fractional Laplacian problems on nonuniform grids, with proven solvability, efficiency improvements, and validated convergence.

Contribution

It develops a novel fast collocation scheme for fractional Laplacian equations on nonuniform grids, including a modified scheme for uniform grids, with rigorous solvability and error analysis.

Findings

Method is uniquely solvable on nonuniform grids for α in (0,1)

Efficient implementation with fast matrix-vector multiplication reduces computational cost

Numerical experiments confirm convergence rates and efficiency

Abstract

We propose a fast collocation method based on Krylov subspace iterative solver on general nonuniform grids for the fractional Laplacian problem, in which the fractional operator is presented in a singular integral formulation. The method is proved to be uniquely solvable on general nonuniform grids for $\alpha\in(0,1)$, provided that the sum-of-exponentials (SOE) approximation is sufficiently accurate. In addition, a modified scheme is developed and proved to be uniquely solvable on uniform grids for $\alpha\in(0,2)$. Efficient implementation of the proposed fast collocation schemes based on fast matrix-vector multiplication is carefully discussed, in terms of computational complexity and memory requirement. To further improve computational efficiency, a banded preconditioner is incorporated into the Krylov subspace iterative solver. A rigorous maximum-norm error analysis for $\alpha\in(0,1)$ is presented on specific graded grids, which shows that the convergence order depends on the grading parameter. Numerical experiments validate the predicted convergence and demonstrate the efficiency of the fast collocation schemes.