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. 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：本文针对涉及对数拉普拉斯算子（作为分数阶拉普拉斯算子在指数$s\to 0^+$时的一阶展开式出现的伪微分算子）的一维Dirichlet问题，提出有限元方法（FEM）的数值分析。我们的分析揭示了该背景下的新现象；特别地，利用近期获得的正则性结果，我们通过采用适当的对数加权空间，证明了严格的误差估计并在能量范数下给出了对数阶收敛率。数值证据表明此类收敛率无法进一步提升。此外，我们证明了对数问题的刚度矩阵可通过分数阶刚度矩阵在$s=0$处的导数获得。最后，我们研究了离散特征值问题及其向连续问题收敛的关系。