Conforming elliptic finite-element reduced gradients have an energy-norm error budget

Apply [F2] to an exact conforming Galerkin state and its adjoint, or [F3] to each conforming perturbed equation, using the transposed form $a^{†}(v,w)=a(w,v)$ for the adjoint. Transposition preserves boundedness and coercivity constants. Under [A1], the resulting two bounds are $E_u,E_p$, with all applicable form and load defects, including geometry and quadrature contributions, retained as specified by [F3].

Choose a basis of $V_h$. Coercivity makes the discrete state matrix $K_h(\theta )$ invertible, and its entries and the load coordinates are $C^1$. Uniform coercivity makes $K_h(\theta +z{)}^{-1}$ locally bounded. The identity $K_h(\theta +z{)}^{-1}-K_h(\theta {)}^{-1}=-K_h(\theta +z{)}^{-1}(K_h(\theta +z)-K_h(\theta ))K_h(\theta {)}^{-1}$ first gives continuity of the inverse. Using $K_h(\theta +z)-K_h(\theta )=D{K}_h(\theta )[z]+o(\mathrm{\parallel }z{\mathrm{\parallel }}_2)$ in that identity gives $D(K_h^{-1})(\theta )[z]=-K_h(\theta {)}^{-1}D{K}_h(\theta )[z]K_h(\theta {)}^{-1}$; this derivative is continuous, so $K_h^{-1}$ and hence $u_h(\theta )$ are $C^1$. Differentiating the state equation gives $a_{\theta ,h}(w_h,v_h)={\mathrm{\ell }}_{\theta ,h}^{\mathrm{\prime }}[\eta ](v_h)-a_{\theta ,h}^{\mathrm{\prime }}[\eta ](u_h,v_h)$. By [F4], $D{j}_h[\eta ]=J_{\theta ,h}^{\mathrm{\prime }}[\eta ]+D_u{J}_h[w_h]$; use the discrete adjoint equation with $v_h=w_h$, then the differentiated state equation with $v_h=p_h$, to obtain the formula for $g_h$ in the Statement. The continuous formula is supplied by [F1].

Fix $\eta$ with $\mathrm{\parallel }\eta {\mathrm{\parallel }}_2\le 1$. Subtract the continuous and discrete formulas. The direct-objective term is bounded by ${\epsilon }_J$. Add and subtract ${\mathrm{\ell }}_{\theta }^{\mathrm{\prime }}[\eta ](p_h)$; then $\mathrm{\mid }{\mathrm{\ell }}_{\theta ,h}^{\mathrm{\prime }}[\eta ](p_h)-{\mathrm{\ell }}_{\theta }^{\mathrm{\prime }}[\eta ](p)\mathrm{\mid }\le {\epsilon }_{\mathrm{\ell }}\mathrm{\parallel }{p}_h{\mathrm{\parallel }}_V+M_{\mathrm{\ell }}\mathrm{\parallel }{p}_h-p{\mathrm{\parallel }}_V\le {\epsilon }_{\mathrm{\ell }}{P}_{\ast }+M_{\mathrm{\ell }}{E}_p$.

Add and subtract $a_{\theta }^{\mathrm{\prime }}[\eta ](u_h,p_h)$. The derivative-form data defect contributes at most ${\epsilon }_a{U}_{\ast }{P}_{\ast }$. The remaining perturbation equals $a_{\theta }^{\mathrm{\prime }}[\eta ](u-u_h,p_h)+a_{\theta }^{\mathrm{\prime }}[\eta ](u,p-p_h)$ up to an overall sign, so $\mathrm{\mid }{a}_{\theta }^{\mathrm{\prime }}[\eta ](u-u_h,p_h)+a_{\theta }^{\mathrm{\prime }}[\eta ](u,p-p_h)\mathrm{\mid }\le M_a(E_u{P}_{\ast }+U_{\ast }{E}_p)\mathrm{.}$ Thus for every unit $\eta$, $\mathrm{\mid }{g}_h[\eta ]-g[\eta ]\mathrm{\mid }$ is bounded by the displayed constant.

Taking the supremum over all $\mathrm{\parallel }\eta {\mathrm{\parallel }}_2\le 1$ gives the asserted dual norm bound. The separate $\epsilon$ terms show why state error alone does not certify a reduced-gradient error.