Cellwise geometry tensor gaps bound form error and stability loss

Set $v=U\circ {\mathrm{\Phi }}_h$ and $w=W\circ {\mathrm{\Phi }}_h$. By [F2], $\mathrm{\nabla }U\circ {\mathrm{\Phi }}_h=D{\mathrm{\Phi }}_h^{-T}\mathrm{\nabla }v$ and likewise for $W$. Under the Countable Choice hypothesis [A3], apply [F3] to the nonnegative measurable energy density $\mathrm{\nabla }{U}^{T}A\mathrm{\nabla }U$ to obtain $a(U,U)=a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}(v,v)$; the same applies to $W$ and $U+W$. The physical energies are finite since $A⪯\mathrm{\Lambda }I$ and all three physical gradients are in $L^2$. Their nonnegative pulled-back energy densities therefore have finite integrals. The pointwise polarization identity expresses the pulled-back cross density as half the difference of these three integrable nonnegative densities, hence makes that cross density absolutely integrable. The physical cross density is integrable by the same argument. Since $A$ and $B_h$ are symmetric, the real polarization identity $2a(U,W)=a(U+W,U+W)-a(U,U)-a(W,W)$ yields $a(U,W)=a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}(v,w)$.

For every vector $\xi$, [A1] and $A⪰\lambda I$ imply ${\xi }^T{B}_h\xi =J_h(D{\mathrm{\Phi }}_h^{-T}\xi {)}^{T}A({\mathrm{\Phi }}_h)(D{\mathrm{\Phi }}_h^{-T}\xi )\ge j_{-}\lambda \mathrm{\parallel }D{\mathrm{\Phi }}_h^{-T}\xi {\mathrm{\parallel }}^2\ge (j_{-}\lambda \mathrm{/}{L}^2)\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$. Thus $B_h⪰\beta I$. The change-of-variables supplier [F3] carries the null exceptional set in the a.e. coefficient bounds to a null set under ${\mathrm{\Phi }}_h$, so these estimates hold almost everywhere after pullback. These same bounds also make the lifted forms finite on every $Z_h$ vector with an $L^2$ gradient: all singular values of $D{\mathrm{\Phi }}_h$ are at most $L$ and their product is $J_h\ge j_{-}$, so $\mathrm{\parallel }D{\mathrm{\Phi }}_h^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le L^{d-1}\mathrm{/}{j}_{-}$ almost everywhere; also $J_h\le L^d$. Hence $\mathrm{\parallel }{B}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }},\mathrm{o}\mathrm{p}}\le \mathrm{\Lambda }{L}^{3d-2}\mathrm{/}{j}_{-}^2$, and the finite cellwise discrepancies make ${\tilde{B}}_h$ essentially bounded as well.

The finite measurable partition and the assumed finite cellwise squared-gradient integrals give global square integrability: apply the measure property [F7] to $\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$ and $\mathrm{\mid }\mathrm{\nabla }w{\mathrm{\mid }}^2$ to obtain $\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2={\sum }_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}^2$ and the analogous identity for $w$. The tensor and gradient entries are measurable, so the defect density $q=\mathrm{\nabla }{v}^T(B_h-{\tilde{B}}_h)\mathrm{\nabla }w$ is measurable by [F7]. On each cell $\mathrm{\mid }q\mathrm{\mid }\le {\eta }_{G,K}\mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }w\mathrm{\mid }$ almost everywhere; [F4] makes this an integrable majorant, and the finite sum makes $q$ globally $L^1$. Step 1.1 has established that the exact lifted cross density is $L^1$. Thus the implemented cross density, the exact cross density minus $q$, is also $L^1$. Their finite indicator decompositions over the partition and $L^1$ linearity [F7] justify subtracting and summing cell integrals. The operator-norm estimate and integral Cauchy–Schwarz [F4] then give $\mathrm{\mid }{\int }_K\mathrm{\nabla }{v}^T(B_h-{\tilde{B}}_h)\mathrm{\nabla }w\mathrm{\mid }\le {\eta }_{G,K}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(K)}$. Summing over cells and using ${\eta }_{G,K}\le {\eta }_G$ together with finite Cauchy–Schwarz [F4] proves both displayed form-defect bounds; under [A2], ${\eta }_G\le C_G{h}^s$.

The diagonal estimate in step 2.1 has constant ${\eta }_G$. On $Z_h$, the exact lifted form is coercive with constant $\beta$ by step 1.2 and [F6]. Apply [F5] with $V=V_h=Z_h$ to obtain coercivity of the implemented form with constant at least $\beta -{\eta }_G\ge \beta \mathrm{/}2$.