P1/R3 practical DPG quasi-optimality and residual estimator for Poisson

Assume Countable Choice. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be any bounded connected Lipschitz polygon, including polygons with holes, and let ${\mathcal{T}}_h$ be any finite conforming shape-regular triangular mesh. Use the exact two-dimensional P1/R3 practical family selected in PE30: volume fields ${\sigma }_h{\mathrm{\mid }}_K\in P_1(K{)}^2$ and $u_h{\mathrm{\mid }}_K\in P_1(K)$; compatible global scalar traces of degree at most two on facets, zero on boundary facets; conservative global normal traces of degree at most one; and broken tests $V_r={\prod }_K(P_3(K{)}^2\times P_3(K))$, with the graph inner product inherited from $V$. The local Fortin projection $\mathrm{\Pi }:V\to V_r$ is that of A uniformly bounded Fortin map for the practical P1/R3 DPG family, and $C_{\mathrm{\Pi }}$ is its operator norm bound. For each $f\in L^2(\mathrm{\Omega })$, the continuous ultraweak solution $x$ and practical solution $x_h$ exist uniquely, with $\mathrm{\parallel }x-x_h{\mathrm{\parallel }}_U\le (1+5C_{\mathrm{\Pi }}{C}_{\ast }){\inf }_{z_h\in U_h}\mathrm{\parallel }x-z_h{\mathrm{\parallel }}_U\mathrm{.}$ For every ${\tilde{x}}_h\in U_h$, put $\eta =\mathrm{\parallel }{F}_f-B{\tilde{x}}_h{\mathrm{\parallel }}_{V_r^{\mathrm{\prime }}}$, where $V_r^{\mathrm{\prime }}$ is the dual of $V_r$ with its induced norm, and $\mathrm{osc}(F_f)=\mathrm{\parallel }{F}_f\circ (I-\mathrm{\Pi }){\mathrm{\parallel }}_{V^{\mathrm{\prime }}}$. Then $C_{\ast }^{-2}\mathrm{\parallel }x-{\tilde{x}}_h{\mathrm{\parallel }}_U^2\le {\eta }^2+(C_{\mathrm{\Pi }}\eta +\mathrm{osc}(F_f){)}^2,\eta \le 5\mathrm{\parallel }x-{\tilde{x}}_h{\mathrm{\parallel }}_U,$ and $\mathrm{osc}(F_f)\le C_{\mathrm{o}\mathrm{s}\mathrm{c}}{({\sum }_K{h}_K^2\mathrm{\parallel }f-f_K{\mathrm{\parallel }}_{L^2(K)}^2)}^{1\mathrm{/}2},f_K=\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_{K}f,$ where $h_K=\mathrm{diam}(K)$ and $C_{\mathrm{o}\mathrm{s}\mathrm{c}}$ is the cellwise constant from the Fortin map's mean-preserving local $L^2$ estimate. The constants $C_{\ast }$, $C_{\mathrm{\Pi }}$, and $C_{\mathrm{o}\mathrm{s}\mathrm{c}}$ are independent of mesh size for the fixed domain and shape-regular family. The practical corollary is two-dimensional; the continuous graph theorem also covers dimension three.