Approximating and preconditioning the stiffness matrix in the GoFD approximation of the fractional Laplacian

TL;DR

This paper analyzes how different approximations of the dense stiffness matrix in the finite difference method for the fractional Laplacian affect overall accuracy, computational efficiency, and preconditioning, with numerical validation in multiple dimensions.

Contribution

It compares four approximations of the stiffness matrix and introduces two preconditioners, enhancing understanding of their impact on the GoFD method for fractional Laplacian problems.

Findings

Approximation accuracy significantly influences overall solution precision.

Preconditioners based on sparse and circulant matrices improve iterative solver performance.

Numerical experiments validate the effectiveness of the proposed methods in 2D and 3D.

Abstract

In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented.