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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.

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If $f_T$ is the piecewise constant cellwise mean, then indeed $\|f-f_T\|_{L_2} \propto {\cal O}(h)$. Consequently, the term $$ \sum_K h_K^2 \|f-f_T\|^2_{L_2(K)} \le h^2 \|f-f_T\|_{L_2}^2 \le C h^4, $$ which is of higher order than the error term $$ \|\nabla (u-u_N)\|_{L_2}^2 \le C h^2. $$ In other words, the "data oscillation term" can be neglected asymptotically -- though it may be large on a given mesh if the mesh is coarse enough.

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