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

TL;DR

This work analyzes how the accuracy of approximating the dense stiffness matrix $T$ in the grid-overlay finite difference GoFD method affects the overall solution of the fractional Laplacian BVP $(-\Delta)^s u=f$ with $s\in(0,1)$ on arbitrary bounded domains. It introduces four stiffness-matrix approximations—FFT, nuFFT, a spectral, and a modified spectral approach—and examines their accuracy, computational cost, asymptotic decay, and compatibility with preconditioners. The study develops two preconditioners (sparse and circulant) and assesses their effectiveness across the four approximations, with extensive 2D and 3D numerical experiments validating GoFD’s convergence and efficiency. The results provide practical guidance: use FFT for large $s$ and the modified spectral approximation for small $s$, balancing accuracy and speed, while leveraging circulant/ sparse preconditioning to accelerate iterative solves. Overall, the paper demonstrates that careful stiffness-matrix treatment is crucial for reliable and scalable GoFD solutions of fractional Laplacian problems on complex geometries.

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.

Figures (11)

• Figure 1: The error in computing $T$ is plotted as a function of $N_{\text{FD}}$ against the discretization error.
• Figure 2: Example (\ref{['main-example']}) with $s = 0.5$. Convergence histories of GoFD with the stiffness matrix $T$ being computed with FFT with various levels of accuracy (through different values of $M$).
• Figure 3: The decay of $T_{\vec{p}}$ as $|\vec{p}| \to \infty$ for $s = 0.25$, 0.50, and 0.75. The reference line is $T_{\vec{p}} = |\vec{p}|^{-(d+2 s)}$.
• Figure 4: The decay of $\tilde{T}_{\vec{p}}$ as $|\vec{p}| \to \infty$ for $s = 0.25$, 0.50, and 0.75. The reference lines are $T_{\vec{p}} = |\vec{p}|^{-(d+2 s)}$ (for 1D) and $T_{\vec{p}} = |\vec{p}|^{-(d+1)/2}$ (for 2D and 3D).
• Figure 5: The decay of $\tilde{\tilde{T}}_{\vec{p}}$ as $|\vec{p}| \to \infty$ for $s = 0.25$, 0.50, and 0.75. The reference line is $T_{\vec{p}} = |\vec{p}|^{-(d+1)/2}$ (for 2D and 3D).

Theorems & Definitions (5)

• Theorem 2.1: HS-2024-GoFD
• Proposition 1
• proof
• Proposition 2
• proof