H^2(\Omega)$. Then: $${\left\lVert u_\epsilon - u_0 \right\rVert}_{0,\Omega} \leq C\epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega}.$$ Here the constant $C$ depends on $\Omega,\kappa_1,\kappa_2,\Delta$

Since $\Gamma_D \neq \varnothing$, by Poincaré inequality, ${\left\lVert u_\epsilon-\tilde{u}_\epsilon \right\rVert}_{0,\Omega} \leq C(\Omega) {\left\lVert u_\epsilon-\tilde{u}_\epsilon \right\rVert}_V \leq C\epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega}$. It is easy to show ${\left\lVert \tilde{u}_\epsilon - u_0 \right\rVert}_{0,\Omega}\leq C(\Omega) \epsilon {\left\lVert u_0 \right\rVert}_{1,\Omega}$. Then the triangle relation tells us ${\left\lVert u_\epsilon - u_0 \right\rVert}_{0,\Omega} \leq C\epsilon^{1/2}{\left\lVert u_0 \right\rVert}_{2,\Omega}$.

This $L^2$ convergence rate presented here is not optimal while We conjecture that best results is ${\left\lVert u_\epsilon-u_0 \right\rVert}_{0,\Omega} \leq C\epsilon{\left\lVert u_0 \right\rVert}_{2,\Omega}$ (see [@Shen2016]), and \[sec:experiments\] supports this claim. However, gradient information is far more valuable in application, this is why we mainly consider norm ${\left\lvert \cdot \right\rvert}_{1,\Omega}$ or ${\left\lVert \nabla \cdot \right\rVert}_{0,\Omega}$.

A Special Case—Robin Problem {#sec:robin}
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The assumptions \[ass:B1\]-\[ass:C\] are relatively general to cover considerable situations. For example, let us consider a simplified version of \[eq:contact problem hemiform\]:

\[pp:simplified contact problem\] Let $\mathrm{B}(x)$ be a function on ${\mathbb{R}}$ satisfying uniform Lipschitz condition, that is ${\left\lvert \mathrm{B}(x)-\mathrm{B}(y) \right\rvert} \leq L_\mathrm{B} {\left\lvert x-y \right\rvert} \forall x, y \in {\mathbb{R}}$, and the other notations are defined as previous. Then the following hemivariational inequality (variational equality) is solvable: $$\label{eq:simplified contact problem}
