English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present the numerical analysis of a finite element method (FEM) forone-dimensional Dirichlet problems involving the logarithmic Laplacian (thepseudo-differential operator that appears as a first-order expansion of thefractional Laplacian as the exponent $s\to 0^+$). Our analysis exhibits newphenomena in this setting; in particular, using recently obtained regularityresults, we prove rigorous error estimates and provide a logarithmic order ofconvergence 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 beobtained as the derivative of the fractional stiffness matrix evaluated at$s=0$. Lastly, we investigate the relationship between the discrete eigenvalueproblem and its convergence to the continuous one.

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