This paper is concerned with error estimates for the piecewise linear finite element approximation of the two-dimensional scalar Signorini problem on a convex polygonal domain $$\varOmega $$ Ω . Using a Céa-type lemma, a supercloseness result, and a non-standard duality argument, we prove $$W^{1,p}(\varOmega )$$ W 1 , p ( Ω ) -, $$L^\infty (\varOmega )$$ L ∞ ( Ω ) -, $$W^{1,\infty }(\varOmega )$$ W 1 , ∞ ( Ω ) -, and $$H^{1/2}(\partial \varOmega )$$ H 1 / 2 ( ∂ Ω ) -error estimates under reasonable assumptions on the regularity of the exact solution and $$L^p(\varOmega )$$ L p ( Ω ) -error estimates under comparatively mild assumptions on the involved contact sets. The obtained orders of convergence turn out to be optimal for problems with essentially bounded right-hand sides. Our results are accompanied by numerical experiments which confirm the theoretical findings.

中文翻译：

---

二维标量 Signorini 问题的非能量范数中的有限元误差估计

本文涉及凸多边形域 $$\varOmega $$ Ω 上二维标量 Signorini 问题的分段线性有限元近似的误差估计。使用 Céa 型引理、超接近性结果和非标准对偶论证，我们证明 $$W^{1,p}(\varOmega )$$ W 1 , p ( Ω ) -, $$L^\ infty (\varOmega )$$ L ∞ ( Ω ) -, $$W^{1,\infty }(\varOmega )$$ W 1 , ∞ ( Ω ) - 和 $$H^{1/2}( \partial \varOmega )$$ H 1 / 2 ( ∂ Ω ) - 在精确解的规律性的合理假设下的误差估计和 $$L^p(\varOmega )$$ L p ( Ω ) - 在比较下的误差估计对所涉及的接触集的温和假设。所获得的收敛阶数对于右手边基本上有界的问题来说是最优的。