Ultraweak Poisson DPG spaces and plan-consistent bilinear form

Assume Countable Choice. Let $d=2$ or $3$, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ bounded connected Lipschitz polyhedral and ${\mathcal{T}}_h$ a finite conforming shape-regular simplicial partition; all fields are real. For a load $f\in L^2(\mathrm{\Omega };\mathbb{R})$, put $H(div,{\mathcal{T}}_h)={\prod }_{K}H(div;K)$ and $H^1({\mathcal{T}}_h)={\prod }_K{H}^1(K)$, with squared broken graph norm ${\sum }_K(\mathrm{\parallel }\tau {\mathrm{\parallel }}_2^2+\mathrm{\parallel }div\tau {\mathrm{\parallel }}_2^2+\mathrm{\parallel }v{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2)$. Let $\widehat{U}$ be the elementwise trace range of $H_0^1(\mathrm{\Omega })$ and ${\widehat{Q}}_n$ the elementwise weak normal trace range of $H(div;\mathrm{\Omega })$, each with global lift infimum norm; range membership is compatibility and normal traces are conservative across interior facets. Set $U=L^2(\mathrm{\Omega }{)}^d\times L^2(\mathrm{\Omega })\times \widehat{U}\times {\widehat{Q}}_n$, $V=H(div,{\mathcal{T}}_h)\times H^1({\mathcal{T}}_h)$. For $x=(\sigma ,u,\widehat{u},{\widehat{\sigma }}_n)$ and $y=(\tau ,v)$, define $b(x,y)=(\sigma ,\tau )-(u,di{v}_h\tau )+⟨\widehat{u},\tau \cdot n⟩-(\sigma ,{\mathrm{\nabla }}_{h}v)+⟨{\widehat{\sigma }}_n,v⟩,F_f(y)=(f,v)\mathrm{.}$ Pairings sum over cell boundaries and use the cell-outward normal. The sign convention is $\sigma =-\mathrm{\nabla }u$, $div\sigma =f$.

Countable Choice is declared for Sobolev traces. The planar rotation identity is proved locally in Planar rotation carries H(div) normal traces to tangential traces; no 3D H(curl) trace result is imported.