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

Area of Science:

• Numerical analysis
• Partial differential equations
• Computational mathematics

Background:

• The classical obstacle problem is a type of variational inequality with applications in various fields.
• Finite-element methods are widely used for approximating solutions to differential equations.
• Error analysis is crucial for understanding the accuracy of numerical methods.

Purpose of the Study:

• To review and analyze the finite-element approximation of the classical obstacle problem.
• To derive error estimates for both the solution and the free boundary.
• To present an optimal error analysis for the thin obstacle problem and discuss convergence rates for localized fractional Laplacian obstacle problems.

Main Methods:

• Finite-element approximation
• Energy and max-norms
• Error estimation
• Regularity results
• Localization techniques

Main Results:

• Error estimates for the solution and free boundary in the classical obstacle problem.
• Optimal error analysis for the thin obstacle problem based on recent regularity results.
• Quasi-optimal convergence rates for the localized obstacle problem involving the fractional Laplacian.

Conclusions:

• The finite-element method provides accurate approximations for obstacle problems.
• Recent regularity results enable optimal error analysis for the thin obstacle problem.
• Localization techniques are effective for analyzing fractional Laplacian obstacle problems, achieving quasi-optimal convergence.