In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in $L^{p}$, for $p \in (4,\infty)$ in two spatial dimensions. This new analysis treats the positive and negative parts of the discretisation error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in $W^{2,(4 - \varepsilon)/3}$ for any $\varepsilon \ll 1$. We summarise extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.

翻译：本文研究了Signorini问题的一致分片线性有限元逼近的后验误差界。我们在二维空间中针对 $p \in (4,\infty)$ 证明了 $L^{p}$ 范数下残差型的新型严格后验估计。该新分析将离散误差的正部和负部分别处理，需要一种新颖的符号与界保持插值算子，该算子被证明具有最优逼近性质。这些估计依赖于问题在 $W^{2,(4 - \varepsilon)/3}$ 空间中对任意 $\varepsilon \ll 1$ 成立的尖锐对偶稳定性结果。我们总结了旨在测试估计子鲁棒性的大量数值实验，以验证该理论。