Geometry lift and transformed elliptic tensor

Assume Countable Choice. Let ${\mathrm{\Omega }}_h,\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, and let ${\mathrm{\Phi }}_h:{\mathrm{\Omega }}_h\to \mathrm{\Omega }$ be a $C^1$ diffeomorphism. Write $J_h(y):=\mathrm{\mid }\det D{\mathrm{\Phi }}_h(y)\mathrm{\mid }>0$. Let $A:\mathrm{\Omega }\to {\mathbb{R}}^{d\times d}$ be Lebesgue measurable, real symmetric almost everywhere, and satisfy $\lambda I⪯A(x)⪯\mathrm{\Lambda }I$ almost everywhere, for constants $0<\lambda \le \mathrm{\Lambda }<\mathrm{\infty }$. The physical exact form on real-valued functions with square-integrable gradients is $a(U,W):={\int }_{\mathrm{\Omega }}\mathrm{\nabla }U(x{)}^{T}A(x)\mathrm{\nabla }W(x)dx$. The pulled-back exact tensor and form are

$B_h(y):=J_h(y)D{\mathrm{\Phi }}_h(y{)}^{-1}A({\mathrm{\Phi }}_h(y))D{\mathrm{\Phi }}_h(y{)}^{-T},a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}(v,w):={\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }v(y{)}^T{B}_h(y)\mathrm{\nabla }w(y)dy\mathrm{.}$

The lifted form is defined for real-valued $v,w$ with measurable gradients and finite pulled-back energies ${\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }{v}^T{B}_h\mathrm{\nabla }v<\mathrm{\infty }$ and ${\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }{w}^T{B}_h\mathrm{\nabla }w<\mathrm{\infty }$. The pointwise Cauchy–Schwarz inequality for the positive tensor $B_h$ bounds the absolute cross density by the product of the square roots of these energy densities, which is integrable by Holder's inequality for integrals, including the endpoint cases. Square-integrability of $\mathrm{\nabla }v,\mathrm{\nabla }w$ alone does not guarantee these energy conditions for an arbitrary diffeomorphism.

The geometry tensor discrepancy of an implemented tensor ${\tilde{B}}_h$ on a cell $K\subset {\mathrm{\Omega }}_h$ is ${\eta }_{G,K}:=\mathrm{\parallel }{B}_h-{\tilde{B}}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K;\mathrm{o}\mathrm{p})}$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ is the Euclidean operator norm. The geometry-lifted form defect is the difference between $a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}$ and the form using ${\tilde{B}}_h$, on pairs for which both displayed cross densities are absolutely integrable. No meshwise approximation rate is part of this definition.