Isogeometric geometry and quadrature defects enter through Strang's bound

By [F7], $V$ is Hilbert in its stated gradient norm; because [F1] makes $F(\widehat{\mathrm{\Omega }})$ bounded, the same Poincare supplier also shows that the physical gradient seminorm is a norm. For any vector $z\in {\mathbb{R}}^d$, [F1] and the map bounds give $z\cdot K_{F}z=J_F\mathrm{\mid }D{F}^{-\mathsf{T}}z{\mathrm{\mid }}^2\ge j_{-}{L}^{-2}\mathrm{\mid }z{\mathrm{\mid }}^2,\mathrm{\parallel }{K}_F{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le j_{+}{L}^2\mathrm{.}$ Indeed, $\mathrm{\mid }D{F}^{-\mathsf{T}}z\mathrm{\mid }\ge \mathrm{\mid }z\mathrm{\mid }\mathrm{/}\mathrm{\parallel }D{F}^{\mathsf{T}}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\ge \mathrm{\mid }z\mathrm{\mid }\mathrm{/}L$ and $\mathrm{\mid }D{F}^{-\mathsf{T}}z\mathrm{\mid }\le L\mathrm{\mid }z\mathrm{\mid }$. Thus ${\widehat{a}}_F$ is coercive and bounded in the gradient norm. The pullback gradient identity and change of variables in [F1] give $\mathrm{\parallel }w\circ F^{-1}{\mathrm{\parallel }}_{\mathrm{\nabla },F}\le \sqrt{j_{+}}L\mathrm{\parallel }w{\mathrm{\parallel }}_V$, so the physical gradient-bounded load pulls back to bounded data. With the exact solution given in the Statement, all exact-form hypotheses of [F4] hold; one may take $M_F=j_{+}{L}^2$.

By [F1], the exact and approximate pulled-back forms use tensors $K_F$ and $K_G$. The form defect in [F3] splits, for any $v_h,w_h$, as ${\widehat{a}}_F(v_h,w_h)-a_h(v_h,w_h)=({\widehat{a}}_F(v_h,w_h)-{\widehat{a}}_G(v_h,w_h))+({\widehat{a}}_G(v_h,w_h)-a_h(v_h,w_h))\mathrm{.}$ Take absolute values, divide by $\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_V$, and take the supremum over nonzero $w_h$; the triangle inequality yields ${\delta }_{a,h}(v_h)\le {\delta }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(v_h)+{\delta }_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}(v_h)$. If $V_h=\{0\}$ all three defects are zero by the Statement and the same comparison is vacuous.

Under the additional map bound in the Statement, [F2] holds for every $w_h$, so taking its supremum gives ${\delta }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(v_h)\le (d{M}^{d+1}+2M^{d+3})\delta \mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V$. This controls only the form's geometry component; it says nothing about the load or quadrature defect.

For NURBS maps $F,G$ satisfying the Statement, apply [F5] on their common knot partition. The resulting reference simplicial mesh is matching, and the cellwise PE17 geometry errors are exactly the same $e_0,e_1$ used here; the homeomorphism $G\circ F^{-1}$ gives the common image topology. The boundary traces of $F$ and $G$ need not be equal.

By [F4], the error is bounded by best approximation plus ${\delta }_{a,h}\mathrm{/}{\alpha }_h$ and ${\delta }_{F,h}\mathrm{/}{\alpha }_h$. Substitute the split from step 1.2, which is exactly the sum of our geometry and quadrature defects, and use ${\delta }_{F,h}={\delta }_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$. Taking the infimum over $v_h$ gives the displayed inequality. Step 1.3 supplies the explicit geometry bound under the additional derivative-map hypotheses.

If the same-basis homogeneous-data hypotheses of [F6] hold and its boundary condition makes $G$ admissible, then its derivative estimate supplies $\delta \le E_1$. Step 1.3 therefore gives ${\delta }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(v_h)\le (d{M}^{d+1}+2M^{d+3})E_1\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V$. In the stated dyadic branch, $E_1\le D{\eta }_b$, so this particular geometry contribution is at most $(d{M}^{d+1}+2M^{d+3})D{\eta }_b\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V$. This does not bound the separate quadrature, load, or discrete-coercivity terms.