# Thin obstacle problem: estimates of the distance to the exact solution

## TL;DR

This paper develops error estimates for elliptic obstacle problems with thin obstacles, providing guaranteed bounds on the distance between approximate and exact solutions without needing the exact solution.

## Contribution

It introduces a method to compute error bounds solely from problem data and approximations, applicable to any solution method for thin obstacle problems.

## Key findings

- Error majorants are guaranteed upper bounds of the approximation error.
- Error estimates vanish if and only if the approximation is exact.
- Efficiency of the error bounds is demonstrated through a numerical example.

## Abstract

We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., it is valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known.