FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension

TL;DR

The paper tackles discretizing the one-dimensional nonlocal Laplacian on nonuniform meshes via a semi-analytic finite element method implemented in the Fourier domain. By deriving explicit integral expressions for the nonlocal stiffness entries, the approach avoids truncation errors and naturally recovers the local FEM stiffness as $\delta\to0$ and the integral fractional Laplacian stiffness as $\delta\to\infty$, establishing asymptotic compatibility. The method supports general interaction kernels $\rho_{\delta}$ and yields structured matrices: pentadiagonal on small horizons and symmetric Toeplitz on uniform grids, with explicit formulas enabling efficient computation. Numerical experiments across smooth and singular kernels, nonlocal BVPs, eigenvalue problems, and a nonlocal Allen–Cahn equation demonstrate accurate convergence, stability, and the expected limiting behavior, while highlighting the benefits of nonuniform meshes for capturing singularities and improving conditioning. The work provides a robust, truncation-free framework for nonlocal FEM in 1D and sets the stage for higher-dimensional extensions and detailed stability analyses.

Abstract

In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius $δ\rightarrow0$ or $δ\rightarrow\infty$ automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.

Figures (11)

• Figure 3.1: Conditioning and the smallest eigenvalue of the stiffness matrix $\boldsymbol{S}_{\delta}$ for $\rho_\delta(s)= \frac{2-\alpha}{\delta^{2-\alpha}} s^{-(1+\alpha)}$ on different meshes with $\delta=0.3$. Condition number: (a) Graded meshes with $\gamma=2$, (b) Geometric meshes with $q = 0.999$, (c) Shishkin meshes with $M = 2N$ and $\eta = 0.2$, (d) The smallest eigenvalue on graded meshes with $\gamma=2$.
• Figure 3.2: The numerical errors for $\rho_\delta(s)= \frac{2-\alpha}{\delta^{2-\alpha}} s^{-(1+\alpha)}$ with different $\alpha \in (-1, 0)$.
• Figure 3.3: The numerical errors of discontinuous solution. (a)-(b): FEM on uniform meshes, (c)-(d): FEM on graded meshes in $L^2$-norm.
• Figure 3.4: Analytical and numerical solutions for $\rho_\delta(s)=\frac{3}{\delta^3}$.
• Figure 3.5: The evolution process of the numerical solutions on graded meshes as $\delta \rightarrow 0$. Top: $\rho_\delta(s)=\frac{3}{\delta^3}$, Bottom: $\rho_{\delta}(s) = \frac{2-\alpha}{\delta^{2-\alpha}}s^{-(1+\alpha)}$ and $\alpha=-0.25$.

Theorems & Definitions (4)

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