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.

中文翻译：

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薄障碍问题：估计到精确解的距离

我们考虑由具有薄障碍物的障碍物类型问题产生的椭圆变分不等式。对于这类问题，我们推导出精确解与满足边界条件且对于障碍条件是可接受的任何函数之间的距离（根据自然能量范数测量）的估计值（即，对于任何近似值，而不管它是通过什么方法找到的）。估计的计算不需要精确解的知识，只使用问题数据和近似值。当且仅当近似值与精确解一致时，估计值提供了有保证的误差上限（误差主项）并消失。在最后一节中，通过一个例子证实了错误majorants的效率，其中精确解是已知的。