I'm trying to prove (uniform) ellipticity of an approximated bilinear form and I've encountered a "problem of approximation".
I posses the quadrature rule $\int_{T'} f'dx'\simeq \sum_i f'(x_i')w'_i$ on the triangle $T'$, which is exact on polynomials of degree $\leq 2k-2$ and has positive weights. Let $T$ be another triangle, and let $\phi: T'\rightarrow T$ be an affine mapping mapping every vertex of $T'$ to a vertex of $T$.
For $u \in \mathbb{P}_{\leq k}(T)$ and $u=0$ on $\partial T$, define $\Delta:=\sum_i w_i'\left [|(\nabla u) \circ\phi|^2|\det\nabla \phi|\right ](x_i')$.
Now, $|(\nabla u) \circ\phi|^2|\det\nabla \phi|$ is for sure a polynomial of degree $\leq 2k-2$ being $\phi$ affine, so that we can write $\Delta= \int_{T'}|(\nabla u) \circ\phi|^2|\det\nabla \phi|dx'$, thanks to the hypothesis of exactness. And now, thanks to a change of variables and the Poincaré inequality, $\Delta=\int_T |\nabla u|^2dx \geq ||u||_{H^1(T)}$.
Problem. Where did I need the quadrature weights to be positive?
In proofs of this kind you do not need positivity of weights at all (in exact arithmetics in other words). However, with non-positive weights the quadrature formula can be numerically unstable, i.e., it can amplify errors, which are always present due to rounding. Consequently, all quadrature rules are preferably chosen to have positive weights. Theory relevant to this problem is very briefly and nicely explained in The Concept of Stability in Numerical Mathematics, Hackbusch W., Springer, 2014, page 25-27.