Problem
Galerkin orthogonality is
but I am not sure if it is in the right form.
How can you use this orthogonality here? I think I should expand the last inequality first somehow.
2026-03-25 09:49:26.1774432166
Galerkin Orthogonality in this FEM?
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\begin{align} \left\| u -u_{h}\right\|_{E}^{2} &= a(u-v_{h} + v_{h} -u_{h}, u-u_{h}) \\ &= a(u-v_{h}, u - u_{h}) + a(v_{h}-u_{h}, u-u_{h}) \\ &= a(u-v_{h}, u - u_{h}) \\ &\le \left\| u -v_{h}\right\|_{E}\left\| u-u_{h}\right\|_{E} \end{align} $a(v_{h}-u_{h}, u-u_{h}) = 0$ by ''Galerkin orthogonality'', as $v_{h} -u_{h} \in V_{h}$.
In particular, \begin{align} \left\| u -u_{h}\right\|_{E} \le \left\| u -v_{h}\right\|_{E} \end{align} for all $v_{h} \in V_{h}$, which implies the conclusion.