The following problem is given:
$$ - \bigtriangleup u = f, \quad \partial_n u = g \quad \text{on } \Gamma_N, \quad u = 0 \quad \text{on } \Gamma_D $$
with $\Gamma_N$ or $\Gamma_D$ denoting the Neumann/Dirichlet boundary.
I am currently having a look at a residual error estimator using the Clément-Interpolant.
There was a theorem stating that the error
$$ \| \nabla (u -u_N) \|_{L^2(\Omega)} \leq C \, \text{EST} = C \, \sqrt{\sum_{K \in T} \eta_K^2},$$
where $\text{EST}$ is an expression that can be calculated with known functions (only dependent on $u_N$, $f$ and $g$). The $\eta_K$ are therefore elementwise indicators for the refinement stategy for each element $K$ in the triangulation $T$.
The next theorem tries to give estimates of $\text{EST}$:
$$ \text{EST}^2 \leq C \left(\| \nabla (u -u_N) \|^2_{L^2(\Omega)} + \sum_{K \in T} h_K^2 \| f - f_T \|^2_{L^2(K)} + \sum_{e \in E_N }h_e \| g - g_T \|^2_{L^2(e)}\right)$$
with $f_T$ the averaging over the triangle $K$, namely $f_T = \frac{1}{|K|} \, \int_{|K|} f$, and $g_T$ the averaging over the edge $e$. $E_N$ are the Neumann edges.
Afterwards, there is a small remark saying that we can hope that the errors of the $\text{EST}^2$-estimate behave approximately like the first term.
The justification of that should be the following: Given a uniform grid, and $f$, $g$ piecewise smooth (like: $f|_K \in H^1(K)$ for all $K$ and $\sum_K \| f \|_{H^1(K)}$ and same for $g$ on the Neumann edge).
Then the terms with $\| f - f_T \|^2_{L^2(K)}$ and $\| g - g_T \|^2_{L^2(e)}$ in the estimate should be of the size $\mathcal{O}({h^3})$ but for the exact solution $u \in H^2$, we expect $\mathcal{O}({h^2})$.
Can you help me to understand, why we are expecting these orders stated in the block?
The $\mathcal{O}({h^2})$ could be due to Poincaré-inequality and changes to the reference element. But I cannot really see how it works.
