Fourier spectral method for nonlocal equations on bounded domains

Ilyas Mustapha; Bacim Alali; Nathan Albin

doi:10.3934/acse.2025022

Article Contents

December 2025, Volume 6, 25-44. Doi: 10.3934/acse.2025022

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Fourier spectral method for nonlocal equations on bounded domains

1.
Department of Mathematics, Kansas State University, Manhattan, KS, 66506, United States

^*Corresponding author: Ilyas Mustapha
^*Corresponding author: Ilyas Mustapha

Received: August 2025

Revised: October 2025

Published online: November 17, 2025

This project is based upon work supported by the National Science Foundation under Grant No. 2108588.

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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 in [1] for periodic domains by incorporating the two-dimensional Fourier continuation (2D-FC) algorithm introduced in [7]. 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.

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Mathematics Subject Classification: Primary: 45A05, 74S25; Secondary: 45K05.

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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.

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Figure 2. Geometric constructions of the 2D-FC procedure, with reference to the regions $ V^- $ and $ V^+ $.

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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.

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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.

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Figure 5. Convergence study for the 2D-FC approximation of the function $ f $ in (6) 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 $.

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Figure 6. Demonstration of the 2D-FC expansion of a nonperiodic function defined on a 2D domain in the settings of Example 2. (a) the original nonperiodic function $ g $. The green curve indicates the boundary of the domain. (b) the 2D-FC expansion of the function $ g $.

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Figure 7. The collar region.

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Figure 8. Simulation results in Example 5 with a fixed $ \delta = 0.4 $ and three choices of $ \beta $.

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Figure 9. Simulation results in Example 6 with a fixed $ \delta = 0.2 $ and four different values of $ \beta $. We observe that the magnitude of the jump at the $ \partial \Omega $ decreases as $ \beta $ increases (see Table 5).

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Figure 10. Simulation results in Example 7 with a fixed $ \delta = 0.5 $ and four values of $ \beta $.

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Figure 11. Simulation results in Example 8 with fixed $ \delta = 0.2 $ and two different values of $ \beta $: 4.0 (first row) and 1.5 (second row) at four different time instances.

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Figure 12. Simulation results in Example 9 with fixed $ \delta = 0.2 $ and two different values of $ \beta $: 4.0 (first row) and 1.0 (second row) at four different time instances.

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Table 1. The multipliers of the nonlocal operator $ \mathcal{L}^{\delta, \beta} $ applied on the function $ u = \sin(2\pi r_1 x)\sin(2\pi r_2 y) $, where $ r_1 = 10.6418 $ and $ r_2 = 12.6418 $, with a fixed $ \delta = 0.4 $ and three different values of $ \beta $.

$ \delta $	$ \overset{\beta = 1.2}{ m^{\delta, \beta}} $	$ \overset{\beta=2.0}{ m^{\delta, \beta}} $	$ \overset{\beta=2.5}{ m^{\delta, \beta}} $
$ 0.4 $	$ -82.87098585883194 $	$ -180.5053934013443 $	$ -387.0397711705603 $

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Table 2. Convergence of the 2D-FC based solution of the nonlocal Poisson equation (9) in the settings of Example 3 with fixed $ \delta = 0.4 $, fixed $ M = d+1 $, and two different values of $ d $.

$ d $	$ h $	$ \overset{\beta=1.2}{\varepsilon_2} $	Order	$ \overset{\beta=2.0}{\varepsilon_2} $	Order	$ \overset{\beta=2.5}{\varepsilon_2} $	Order
4	$ 2.00\cdot 10^{-2} $	$ 1.29\cdot 10^{-04} $	—–	$ 3.56\cdot 10^{-04} $	—–	$ 8.97\cdot 10^{-04} $	—–
	$ 1.00\cdot 10^{-2} $	$ 2.00\cdot 10^{-06} $	$ 6.0 $	$ 4.10\cdot 10^{-06} $	$ 6.4 $	$ 1.09\cdot 10^{-05} $	$ 6.4 $
	$ 5.00\cdot 10^{-3} $	$ 2.65\cdot 10^{-08} $	$ 6.2 $	$ 7.75\cdot 10^{-08} $	$ 5.7 $	$ 2.32\cdot 10^{-07} $	$ 5.6 $
	$ 2.50\cdot 10^{-3} $	$ 1.44\cdot 10^{-10} $	$ 7.5 $	$ 2.19\cdot 10^{-10} $	$ 8.5 $	$ 1.61\cdot 10^{-09} $	$ 7.2 $
	$ 1.25\cdot 10^{-3} $	$ 1.62\cdot 10^{-12} $	$ 6.5 $	$ 4.79\cdot 10^{-12} $	$ 5.5 $	$ 2.24\cdot 10^{-11} $	$ 6.2 $
5	$ 2.00\cdot 10^{-2} $	$ 6.65\cdot 10^{-04} $	—–	$ 2.09\cdot 10^{-03} $	—–	$ 5.11\cdot 10^{-03} $	—–
	$ 1.00\cdot 10^{-2} $	$ 6.66\cdot 10^{-06} $	$ 6.6 $	$ 1.71\cdot 10^{-05} $	$ 6.9 $	$ 4.99\cdot 10^{-05} $	$ 6.7 $
	$ 5.00\cdot 10^{-3} $	$ 4.94\cdot 10^{-08} $	$ 7.1 $	$ 1.39\cdot 10^{-07} $	$ 6.9 $	$ 4.17\cdot 10^{-07} $	$ 6.9 $
	$ 2.50\cdot 10^{-3} $	$ 2.21\cdot 10^{-10} $	$ 7.8 $	$ 5.84\cdot 10^{-10} $	$ 7.9 $	$ 2.63\cdot 10^{-09} $	$ 7.3 $
	$ 1.25\cdot 10^{-3} $	$ 1.31\cdot 10^{-12} $	$ 7.4 $	$ 4.41\cdot 10^{-12} $	$ 7.0 $	$ 2.06\cdot 10^{-11} $	$ 7.0 $

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Table 3. The multipliers of the nonlocal operator $ \mathcal{L}^{\delta, \beta} $ applied on the function $ u = \sin(2\pi r_1 x)\sin(2\pi r_2 y) $, where $ r_1 = r_2 = 15.6455 $, with a fixed $ \delta = 0.3 $ and three different values of $ \beta $.

$ \delta $	$ \overset{\beta = 1.0}{ m^{\delta, \beta}} $	$ \overset{\beta=2.0}{ m^{\delta, \beta}} $	$ \overset{\beta=2.5}{ m^{\delta, \beta}} $
$ 0.3 $	$ -130.16228859689554 $	$ -321.3202730766787 $	$ -689.8419741563309 $

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Table 4. Convergence of the 2D-FC based solution of the nonlocal diffusion equation (12) in the settings of Example 4 with fixed $ \delta = 0.3 $, fixed $ M = d+1 $, two different values of $ d $: $ 4 $ and $ 5 $, and three choices of $ \beta $: $ 1.0 $, $ 2.0 $, and $ 2.5 $.

$ d $	$ h $	$ \overset{\beta=1.0}{\varepsilon_2} $	Order	$ \overset{\beta=2.0}{\varepsilon_2} $	Order	$ \overset{\beta=2.5}{\varepsilon_2} $	Order
4	$ 2.00\cdot 10^{-2} $	$ 1.13\cdot 10^{-05} $	—–	$ 5.33\cdot 10^{-05} $	—–	$ 2.48\cdot 10^{-04} $	—–
	$ 1.00\cdot 10^{-2} $	$ 1.51\cdot 10^{-07} $	$ 6.2 $	$ 9.45\cdot 10^{-07} $	$ 5.8 $	$ 6.81\cdot 10^{-06} $	$ 5.2 $
	$ 5.00\cdot 10^{-3} $	$ 2.07\cdot 10^{-09} $	$ 6.2 $	$ 1.48\cdot 10^{-08} $	$ 6.0 $	$ 1.45\cdot 10^{-07} $	$ 5.6 $
	$ 2.50\cdot 10^{-3} $	$ 1.87\cdot 10^{-11} $	$ 6.8 $	$ 9.21\cdot 10^{-11} $	$ 7.3 $	$ 1.22\cdot 10^{-09} $	$ 6.9 $
	$ 1.25\cdot 10^{-3} $	$ 2.43\cdot 10^{-13} $	$ 6.3 $	$ 1.29\cdot 10^{-12} $	$ 6.2 $	$ 2.29\cdot 10^{-11} $	$ 5.7 $
5	$ 2.00\cdot 10^{-2} $	$ 7.87\cdot 10^{-05} $	—–	$ 4.10\cdot 10^{-04} $	—–	$ 1.93\cdot 10^{-03} $	—–
	$ 1.00\cdot 10^{-2} $	$ 8.84\cdot 10^{-07} $	$ 6.5 $	$ 5.86\cdot 10^{-06} $	$ 6.1 $	$ 4.16\cdot 10^{-05} $	$ 5.5 $
	$ 5.00\cdot 10^{-3} $	$ 5.53\cdot 10^{-09} $	$ 7.3 $	$ 3.30\cdot 10^{-08} $	$ 7.5 $	$ 3.23\cdot 10^{-07} $	$ 7.0 $
	$ 2.50\cdot 10^{-3} $	$ 2.99\cdot 10^{-11} $	$ 7.5 $	$ 1.87\cdot 10^{-10} $	$ 7.5 $	$ 2.49\cdot 10^{-09} $	$ 7.0 $
	$ 1.25\cdot 10^{-3} $	$ 3.49\cdot 10^{-13} $	$ 6.4 $	$ 1.27\cdot 10^{-12} $	$ 7.2 $	$ 2.15\cdot 10^{-11} $	$ 6.9 $

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Table 5. The minimum and the maximum magnitudes of the jumps at $ \partial \Omega $ in the simulation results presented in Figure 9. We observe that the magnitudes of the jumps decrease as $ \beta $ increases.

	$ \beta = 1.5 $	$ \beta=2.0 $	$ \beta=2.2 $	$ \beta=3.1 $
$ \min $ magnitude	$ 0.0538468 $	$ 0.0286598 $	$ 0.0187264 $	$ 6.24\cdot 10^{-05} $
$ \max $ magnitude	$ 0.0646947 $	$ 0.0454215 $	$ 0.0372901 $	$ 0.0097378 $

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Table 6. The minimum and the maximum magnitudes of the jumps in the interior of $ \Omega $ in the simulation results presented in Figure 10. We observe that the magnitudes of the jumps decrease as $ \beta $ increases.

	$ \beta = 1.2 $	$ \beta=2.0 $	$ \beta=2.5 $	$ \beta=3.1 $
$ \min $ magnitude	$ 1.3589861 $	$ 0.3237825 $	$ 0.0363031 $	$ 1.75\cdot 10^{-05} $
$ \max $ magnitude	$ 1.3681906 $	$ 0.4040726 $	$ 0.0856388 $	$ 0.0067363 $

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