Bounded velocity-observation error for the smooth MINI branch

Assume Countable Choice and a smooth MINI observation interface as specified by the interface definition on this page. In particular, let $D\subset {\mathbb{R}}^d$ be bounded, connected, Lipschitz and polygonal or polyhedral with $d\in \{2,3\}$; use a shape-regular conforming affine simplicial mesh, viscosity $\nu >0$, $f\in L^2(0,T;L^2(D{)}^d)$, the mass-projected initial value, and any actual mixed backward Euler MINI solution. Suppose the exact no-slip solution has $u\in W^{1,\mathrm{\infty }}(0,T;H^2(D{)}^d\cap V_{\sigma })$ and $p\in W^{1,\mathrm{\infty }}(0,T;H^1(D)\cap L_0^2(D))$. Set $M=1+\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}{H}^2}+\mathrm{\parallel }p{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}{H}^1},H=hM+{\tau }_{\max }{M}^2,M_u=\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(0,T;V)},\mathrm{\Lambda }=1+C_D{\nu }^{-3}{M}_u^4\mathrm{.}$ Assume ${\tau }_{\max }\mathrm{\Lambda }\le \frac{1}{2}$. Let $M_{\mathrm{\infty }}$, $M_E$ and $C_{\ast }$ have exactly the values in the smooth MINI velocity-error estimate, namely $M_{\mathrm{\infty }}=(\mathrm{\parallel }u(0){\mathrm{\parallel }}_2^2+C_P(D{)}^2{\nu }^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2{L}^2}^2{)}^{1\mathrm{/}2},M_E={\nu }^{-1\mathrm{/}2}\mathrm{\parallel }u(0){\mathrm{\parallel }}_2+C_P(D){\nu }^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2{L}^2},$ $C_{\ast }=C_{D,\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}}(1+\nu {)}^2(1+{\nu }^{-4})(1+T{)}^2(1+M_{\mathrm{\infty }}+M_E+M{)}^4\exp (2T\mathrm{\Lambda }),$ where $C_D$ depends only on the domain, dimension and trilinear-form constant, and $C_{D,\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}}$ is independent of $h$, the time steps and $\nu$. No mesh/time coupling is imposed beyond the stated time-step restriction. For any sensors ${\mathrm{\ell }}_n\in V^{\ast }$, define $\mathrm{\parallel }\mathrm{\ell }{\mathrm{\parallel }}_{{\mathrm{\ell }}_{\tau }^2(V^{\ast })}=({\sum }_{n=1}^N{\tau }_n\mathrm{\parallel }{\mathrm{\ell }}_n{\mathrm{\parallel }}_{V^{\ast }}^2{)}^{1\mathrm{/}2}$. Then the endpoint-sampled observations of the interface satisfy $\mid {\mathcal{O}}_h(\mathrm{\ell };u_h)-\mathcal{O}(\mathrm{\ell };u)\mid \le {\nu }^{-1\mathrm{/}2}{C}_{\ast }H\mathrm{\parallel }\mathrm{\ell }{\mathrm{\parallel }}_{{\mathrm{\ell }}_{\tau }^2(V^{\ast })}\mathrm{.}$ This is a conditional bound for bounded linear velocity sensors on the smooth branch; it makes no Reynolds-uniform claim.

smooth MINI observation interface

smooth-branch MINI velocity error theorem

Guermond, Chapter 54, §54.2

Boffi, Brezzi, and Fortin, §§8.4.2 and 8.6.1