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 \emph{log}-weighted spaces. Numerical evidence suggests that this type of rate cannot be improved. 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.

翻译：本文针对包含对数拉普拉斯算子（即分数阶拉普拉斯算子指数$s\to 0^+$时一阶展开所出现的伪微分算子）的一维Dirichlet问题，提出有限元法的数值分析。分析揭示了该设定下的新现象：特别地，基于最新获得的正则性结果，我们在合适的对数加权空间中证明了严格的误差估计，并给出了能量范数下的对数收敛阶。数值实验表明此类收敛速率无法进一步改进。此外，我们证明了对数问题的刚度矩阵可通过对分数阶刚度矩阵在$s=0$处求导获得。最后，我们探讨了离散特征值问题及其向连续特征值问题收敛之间的关系。