Fourier Spectral Method for Nonlocal Equations on Bounded Domains

TL;DR

This work develops a Fourier continuation–based spectral solver for two-dimensional nonlocal Poisson and diffusion equations on bounded domains. By representing the nonlocal operator with Fourier multipliers $m^{\delta,\beta}$ and extending nonperiodic data to a periodic domain via the 2D-FC algorithm, the method achieves fast $O(N\log N)$ computations while preserving high-order accuracy for smooth solutions. The authors implement a GMRES-based solver for Poisson and a fourth-order Adams–Bashforth time stepping for diffusion, and they investigate regularity and possible discontinuities under varying kernel exponents $\beta$, providing numerical evidence for conjectured continuity in the interior when $\beta>3$. Numerical results demonstrate strong convergence for smooth cases and reveal how discontinuities depend on $\beta$, with jumps diminishing as kernels become more singular. The framework offers a practical and scalable tool for nonlocal models on bounded domains, with implications for peridynamics and related applications.

Abstract

This work introduces efficient and accurate spectral solvers for nonlocal equations on bounded domains. These spectral solvers exploit the fact that integration in the nonlocal formulation transforms into multiplication in Fourier space and that nonlocality is decoupled from the grid size, allowing fast and accurate solutions to the nonlocal problems. Our approach extends the spectral solvers developed by Alali and Albin (2020) for periodic domains by incorporating the two-dimensional Fourier continuation (2D-FC) algorithm introduced by Bruno and Paul (2022). We evaluate the performance of the proposed methods on two-dimensional nonlocal Poisson and nonlocal diffusion equations defined on bounded domains. While the regularity of solutions to these equations in bounded settings remains an open problem, we conduct numerical experiments to explore this issue, particularly focusing on studying discontinuities.

Figures (12)

• Figure 1: A demonstration of the 1D blending to zero on a smooth but nonperiodic function $f(x)=\sin(15x)\exp(x)$ defined on [0,1]. The black solid points denote the $d$ near-boundary function values used to produce the extension of $f$ denoted by the blue solid points.
• Figure 2: Geometric constructions of the 2D-FC procedure. The interior strip $V^-$ and the exterior strip $V^+$ are defined in subsection \ref{['curvilinear_grid']}.
• Figure 3: Interpolation procedure for evaluating values of $f$ on the set $V_{\theta_p}^-$ for the case $|n_x(\theta_p)|\ge|n_y(\theta_p)|$. The black solid circles indicate the points that define the set $V^-_{\theta_p}$. The known values of $f$ at the $M$ open circles are used to interpolate the function values at the red stars (intersection points of the normal and the vertical grid lines). The values of $f$ at the red points are then used to obtain the values of $f$ at the black points.
• Figure 4: Interpolation procedure for obtaining continuation values of $f$ on $\mathcal{H}\cap V^+$. In (a), the blue-star interpolation points are the perpendicular projections of the Cartesian point $Q$ onto the normal vectors, and the $\tau$'s are the signed perpendicular distances from $Q$ to each normal vector. In (b), the interpolation of continuation values from the curvilinear grids to a point $Q$ on the Cartesian grids.
• Figure 5: Convergence study for the 2D-FC approximation of the function $f$ in \ref{['f_performance']} with three choices of matching point numbers $d$: $d=4$, $5$, and $8$; and three choices of $M$: $M=d+1$, $d+2$, and $d+3$. The integer $N_{1D}$ is the number of spatial grid points in one spatial direction within $\Omega$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Example 1
• Example 2: Graphical illustration of 2D-FC method
• Example 3: Convergence test for the nonlocal Poisson equation
• Example 4: Convergence test for nonlocal diffusion equation
• Example 5
• Example 6
• Example 7
• Conjecture 1