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

2023/11/22 by Víctor Hernández-Santamaría, Sven Jarohs, Hernández-Santamaría, Víctor +5 · 3 citations
Computer Science · Engineering · #35B65 #35S15 #65N15 #65N25 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2311.13079

openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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→ 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.

Cited by

• Sharp embeddings and existence results for Logarithmic p-Laplacian equations with critical growth
• Numerical Approximation of the logarithmic Laplacian via sinc-basis
• The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces

Related

• Finite Elements with weighted bases for the fractional Laplacian
• A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations
• An implementation of hp-FEM for the fractional Laplacian
• On explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes
• A Finite Element approximation of the one-dimensional fractional Poisson equation with applications to numerical control