Adaptive mesh refinement

Remarks

A common marking rule is the Dörfler criterion with a parameter $0 < \theta \le 1$. It takes a set $\mathcal{M}_\ell$ of minimal size with $\sum_{K \in \mathcal{M}_\ell} \eta_K^2 \ge \theta \sum_{K \in \mathcal{T}_\ell} \eta_K^2$. For the Poisson equation, a standard residual indicator combines $h_K^2 \norm{f + \Delta u_\ell}_{L^2(K)}^2$ with the jumps of the normal derivative of $u_\ell$ across the faces of $K$.

Example: on the L-shaped domain $(-1, 1)^2 \setminus [0, 1) \times (-1, 0]$, the solution of $-\Delta u = 0$ with boundary values $r^{2/3} \sin(2\varphi/3)$ is $u = r^{2/3} \sin(2\varphi/3)$ in polar coordinates $(r, \varphi)$ with $0 < \varphi < 3\pi/2$. Its gradient blows up at the corner $0$, the indicators are largest there, and the meshes become graded towards $0$.

Non-example: uniform refinement subdivides every element of $\mathcal{T}_\ell$ in each step. Its refinement does not depend on any indicator, so step 3 is missing.