Finite element error estimates in $$L^2$$ for regularized discrete approximations to the obstacle problem

Finite element error estimates in non-energy norms for the two-dimensional scalar Signorini problem

“…The unique solvability of ( D) for all h > 0 follows from [25, Theorem II-2.1], and the non-positivity of z and the property tr(z) = 0 on Ãh are obtained completely analogously to the proof of Lemma 5.2. The same is the case for the estimate (22). It remains to prove the W 2,(4−ε)/3 -regularity of z and ( 23) for all sufficiently small h > 0.…”

“…However, it is easy to check that only two sets B and combinations of boundary conditions are possible here. We may thus again invoke [19,Theorem 4.3.2.4] and use (22) to deduce that, for every ε ∈ (0, 1/2), we have…”

“…Using exactly the same arguments as in the proof of Lemma 5.2, we may now transform the situation at d i into one of finitely many reference configurations and invoke [19,Theorem 4.3.2.4] as well as (22) to deduce that there exists a constant…”

“…We remark that a notable exception to this rule are L ∞ -estimates which can be established along roughly the same lines as for elliptic PDEs in the case of the Signorini problem (1) by employing the discrete maximum principle or regularization techniques, cf. [4,11,13,15,18,22,28,34,35].…”