I have tried to find out the answer by searching but there is no Wiki page like the introduction to trace inequality. Roughly I know the inverse inequality is something like $\|\nabla v\| \lesssim h \|v\|$ but I really want to know more details, and the reason why it is called "inverse".
2026-03-24 20:47:46.1774385266
What is inverse inequality in functional/numerical analysis or finite element methods?
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Poincare's inequality says:
$$||v||_{L^2(\Omega)} \leq C(\Omega)||\nabla v||_{L^2(\Omega)}~~ \forall v\in H_0^1(\Omega) $$ but for quasi-uniform triangulation, we have $$||\nabla v_h||_{L^2} \leq Ch^{-1}|| v_h||_{L^2}, $$
these kinds of estimates are called inverse estimates for the apparent reason that they provide inverse relation to norms.