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

### Darya E. Apushkinskaya

Universität des Saarlandes, Saarbrücken, Germany and Peoples’ Friendship University of Russia (RUDN), Moscow, Russia### Sergey I. Repin

Russian Acadademy of Sciences, St. Petersburg, Russia and Jyväskylä University, Finland

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## 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., they are 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.

## Cite this article

Darya E. Apushkinskaya, Sergey I. Repin, Thin obstacle problem: Estimates of the distance to the exact solution. Interfaces Free Bound. 20 (2018), no. 4, pp. 511–531

DOI 10.4171/IFB/410