Hello,

Assuming that I have the 2D Poisson problem with pure Neumann b.c., is there a way to obtain a residual type estimator for H1

norm, in order to use it in an adaptive algorithm? Namely, I would like to compute the following quantity for every element K

of the mesh,

\[\eta_{1, K}=\left\{h_K^2\left\|f+\Delta u_h\right\|_K^2+\frac{1}{2} \sum_{S \in \partial K \cap S_{int}} h_S\left\|\left[ \nabla u_h\cdot \vec{n}\right]\right\|_S^2+\sum_{S\in \partial K\cap \partial \Omega} h_S\left\|g- \nabla u_h\cdot \vec{n}\right\|_S^2\right\}^{\frac{1}{2}}.\]

Here $f$ and $g$ are the right hand side on the domain and the boundary, respectively, while $S$ is an edge and $S_{int}$ is the set of all interior edges.

Best regards,

Vex