Accumulated pressure-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, $D\subset {\mathbb{R}}^d$ is bounded, connected, Lipschitz and polygonal or polyhedral with $d\in \{2,3\}$; $T>0$, $\nu >0$, and $f\in L^2(0,T;L^2(D{)}^d)$; the exact no-slip solution has $u\in W^{1,\mathrm{\infty }}(0,T;H^2(D{)}^d\cap V_{\sigma })$ and $p\in W^{1,\mathrm{\infty }}(0,T;H^1(D)\cap Q)$; the MINI meshes are shape-regular conforming affine simplices; the initial value is $u_h^0=P_{Z_h}u(0)$; and any actual mixed backward Euler MINI solution is used. No mesh/time relation is imposed. Impose the time-step restriction ${\tau }_{\max }\mathrm{\Lambda }\le \frac{1}{2}$ of the smooth-branch pressure-primitive theorem. Set $M=1+\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}{H}^2}+\mathrm{\parallel }p{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}{H}^1}$, $H=hM+{\tau }_{\max }{M}^2$, $M_u=\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(0,T;V)}$, $M_E={\nu }^{-1\mathrm{/}2}\mathrm{\parallel }u(0){\mathrm{\parallel }}_2+C_P(D){\nu }^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(0,T;L^2(D{)}^d)}$, and $\mathrm{\Lambda }=1+C_D{\nu }^{-3}{M}_u^4$, where $C_D$ depends only on the domain, dimension and trilinear-form constant. Let $C_{\ast }$ be exactly the explicit viscosity-dependent constant in the smooth MINI velocity-error theorem. Let $\beta >0$ be the uniform MINI inf-sup lower bound, let $M_b=\sqrt{d}$, let $C_I$ be a mean-adjusted continuous P1 pressure interpolation constant satisfying ${\inf }_{q_h\in Q_h}\mathrm{\parallel }q-q_h{\mathrm{\parallel }}_2\le C_{I}h\mathrm{\parallel }q{\mathrm{\parallel }}_{H^1(D)}$ for $q\in H^1(D)\cap Q$, and choose $C_c$ such that $\mathrm{\mid }c(a;b,v)\mathrm{\mid }\le C_c\mathrm{\parallel }a{\mathrm{\parallel }}_V\mathrm{\parallel }b{\mathrm{\parallel }}_V\mathrm{\parallel }v{\mathrm{\parallel }}_V$. Let $C_P(D)$ satisfy $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(D)}\le C_P(D)\mathrm{\parallel }v{\mathrm{\parallel }}_V$ for $v\in V$. With $K=(1+M_b\mathrm{/}\beta )C_{I}T+{\beta }^{-1}(2C_P+[2\nu \sqrt{T}+C_c(M_E+\sqrt{T}{M}_u)]{\nu }^{-1\mathrm{/}2})C_{\ast }+{\beta }^{-1}(2\nu T+2C_{c}T)\mathrm{.}$ For every $\xi \in Q$, every $\eta \in C_c^{\mathrm{\infty }}(0,T)$, and every node $0\le m\le N$, the interface observations satisfy $\mid (P(t_m)-P_h^m,\xi {)}_{L^2}\mid \le KH\mathrm{\parallel }\xi {\mathrm{\parallel }}_Q\mathrm{.}$

$\mid {({\int }_0^T(p(t)-{\bar{p}}_h(t))\eta (t)dt,\xi )}_{L^2}\mid \le (K+1)H\mathrm{\parallel }{\eta }^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^1(0,T)}\mathrm{\parallel }\xi {\mathrm{\parallel }}_Q\mathrm{.}$ These are accumulated-pressure and time-tested pressure bounds on the smooth branch. They do not assert an instantaneous pressure-error rate.

smooth MINI observation interface

smooth MINI pressure-primitive error theorem

smooth MINI velocity error theorem

Guermond, Chapter 54, §54.2

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