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    $\begingroup$ Well, the derivative phisol is almost zero. I get -0.0302489. Since this is numerical method and not exact, may be trying to decrease the mesh spacing or other ways to improve the FEM accuracy could make it closer to zero? $\endgroup$ Commented May 17, 2022 at 7:27
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    $\begingroup$ But I agree that it does look a bit off zero, even for numerical method. But may be trying to improve the FEM accuracy could make this error smaller? Otherwise, it could be a bug. Just speculating. I am sure someone here with more knowledge of FEM in Mathematica will know for sure. $\endgroup$ Commented May 17, 2022 at 7:47
  • $\begingroup$ I'd like to remember everyone that the Neumann condition in FEM is imposed a bit... counterintuitively. Part of that is to interpret $\frac{\partial u}{\partial \nu}$ of a function $u \in H^1(\varOmega)$ on a smooth domain $\varOmega$ as an element in $H^{-1/2}(\partial \varOmega)$. Sloppily speakin, one derivative is lost due to taking the normal derivative, another one is lost due to restriction to the boundary $\partial \varOmega$ which has codimension one. $\endgroup$ Commented May 17, 2022 at 11:53
  • $\begingroup$ So it cannot be expected that $\frac{\partial u_h}{\partial \nu}$ of the discrete solution $u_h$ converges in $L^2$ or $L^\infty$ to $0$ under grid refinement $u_h \to 0$. Nonetheless, $u_h$ can still converge in $H^1$ to the true solution. $\endgroup$ Commented May 17, 2022 at 11:56
  • $\begingroup$ @HenrikSchumacher Oh this is a bit over my head… any recommended reference? $\endgroup$ Commented May 17, 2022 at 12:15