Practical DPG residual is reliable and efficient up to data oscillation

Let $e=x-{\tilde{x}}_h$ and let $ϵ\in V$ represent $F-B{\tilde{x}}_h$ by Riesz. Choose an orthonormal basis $e_1,\dots ,e_N$ of the finite-dimensional polynomial space $V_r$ and set ${ϵ}_h={\sum }_{j=1}^N(ϵ,e_j{)}_V{e}_j$. Then $\delta =ϵ-{ϵ}_h$ is orthogonal to $V_r$, $\eta =\mathrm{\parallel }{ϵ}_h{\mathrm{\parallel }}_V=\mathrm{\parallel }F-B{\tilde{x}}_h{\mathrm{\parallel }}_{V_r^{\mathrm{\prime }}}$, and $\mathrm{\parallel }ϵ{\mathrm{\parallel }}^2={\eta }^2+\mathrm{\parallel }\delta {\mathrm{\parallel }}^2$. Continuous inf-sup gives $\beta \mathrm{\parallel }e\mathrm{\parallel }\le \mathrm{\parallel }ϵ\mathrm{\parallel }$.

Riesz gives $\eta \le \mathrm{\parallel }Be{\mathrm{\parallel }}_{V^{\mathrm{\prime }}}\le \mathrm{\parallel }b\mathrm{\parallel }\mathrm{\parallel }e\mathrm{\parallel }$. For any $z_h\in U_h$, Fortin orthogonality gives $F((I-\mathrm{\Pi })y)=b(x-z_h,(I-\mathrm{\Pi })y)$; take the dual norm and infimum to obtain the oscillation approximation bound.

Since $\mathrm{\Pi }\delta \in V_r$, $\mathrm{\parallel }\delta {\mathrm{\parallel }}^2=(\delta ,\delta -\mathrm{\Pi }\delta )$. Substitute $ϵ-{ϵ}_h$ for $\delta$; Fortin orthogonality kills $b({\tilde{x}}_h,(I-\mathrm{\Pi })\delta )$, leaving $F((I-\mathrm{\Pi })\delta )+({ϵ}_h,\mathrm{\Pi }\delta )$. Hence $\mathrm{\parallel }\delta \mathrm{\parallel }\le \mathrm{osc}(F)+\mathrm{\parallel }\mathrm{\Pi }\mathrm{\parallel }\eta$. Combine with step 1.1 to get the reliability inequality.

For the Poisson load, mean preservation gives ${\int }_K(v-{\mathrm{\Pi }}_3^{grad}v)=0$, so $F_f((I-\mathrm{\Pi })(\tau ,v))={\sum }_K(f-f_K,v-{\mathrm{\Pi }}_3^{grad}v{)}_K$. By [F2], each difference is bounded by $C{h}_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_K$. Cauchy–Schwarz over cells yields the stated data oscillation; it is zero if every $f{\mathrm{\mid }}_K=f_K$.