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arXiv · 2311.13079

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

Víctor Hernández-Santamaría·Sven Jarohs·Alberto Saldaña·Leonard Sinsch
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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.

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Víctor Hernández-Santamaría, Sven Jarohs, Alberto Saldaña, Leonard Sinsch. 2023-11-22. FEM for 1D-problems involving the logarithmic Laplacian: error estimates and numerical implementation. https://arxiv.org/abs/2311.13079

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