FEM for 1D-problems involving the logarithmic Laplacian: error estimates and numerical implementation

TL;DR

This paper develops and analyzes a finite element method for one-dimensional nonlocal Dirichlet problems involving the logarithmic Laplacian $L_\Delta$, establishing a rigorous logarithmic rate of convergence in the energy norm via log-Hölder spaces. A key methodological advance is showing that the logarithmic stiffness matrix is the $s\to0$ derivative of the fractional-Laplacian stiffness, $A_h^L=\partial_s A_h^s|_{s=0}$, which enables straightforward and explicit implementation in 1D. The work also proves stability under a discrete inf-sup condition, analyzes interpolation in weighted log norms, and demonstrates convergence of discrete eigenvalues to the continuous spectrum, with detailed numerical illustrations including explicit solutions and eigenvalue computations. Overall, the results provide a robust, implementable framework for 1D logarithmic nonlocal problems, highlighting unique regularity and spectral features that differ from the fractional Laplacian. The combination of theoretical and numerical insights advances understanding of log-Laplacian problems and supports practical computation in applications where $s\to0^+$ is relevant.

Abstract

We present the numerical analysis of a finite element method (FEM) for one-dimensional Dirichlet problems involving the logarithmic Laplacian (the pseudo-differential operator that appears as a first-order expansion of the fractional Laplacian as the exponent $s\to 0^+$). Our analysis exhibits new phenomena in this setting; in particular, using recently obtained regularity results, we prove rigorous error estimates and provide a logarithmic order of convergence in the energy norm using suitable $\log$-weighted spaces. Moreover, we show that the stiffness matrix of logarithmic problems can be obtained as the derivative of the fractional stiffness matrix evaluated at $s=0$. Lastly, we investigate the relationship between the discrete eigenvalue problem and its convergence to the continuous one.

Figures (7)

• Figure 1: Numerical approximation of the solution to \ref{['t:p:intro']} for different values of $L$.
• Figure 2: The logarithmic Laplacian of $u(x)$ defined in \ref{['udef']}.
• Figure 3: A comparison between $u$ given by \ref{['udef']} and $v_h$ given by \ref{['vdef']} for $N=50$.
• Figure 4: Error curves for $a_h$, $b_h$ (with $K=(-0.9,0.9)$), and $c_h$.
• Figure 5: Numerical approximation of eigenfunctions of $L_{\Delta}$ for $L = 0.1$ (top) and $L = 8$ (bottom).

Theorems & Definitions (55)

• Theorem 1.1: Theorem 1.1 in CS22
• Theorem 1.2
• Proposition 1.3
• Lemma 2.1: Semi-homogeneity
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof