Convergence rates for the classical, thin and fractional elliptic obstacle problems

TL;DR

This paper reviews finite element methods for classical, thin, and fractional obstacle problems, providing error estimates and convergence rates based on recent regularity results.

Contribution

It offers a comprehensive analysis of error estimates and convergence rates for various obstacle problems, including new results for the fractional Laplacian case.

Findings

Optimal error estimates for classical obstacle problem.

Error analysis for thin obstacle problem based on recent regularity.

Quasi-optimal convergence rates for fractional obstacle problem.

Abstract

We review the finite element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results we present an optimal error analysis for the thin obstacle problem. Finally, we discuss the localization of the obstacle problem for the fractional Laplacian and prove quasi-optimal convergence rates.