Smooth MINI observation interface

Assume the Axiom of Countable Choice. Let $d\in \{2,3\}$, let $D\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz polygonal or polyhedral open set, let $T>0$ and $\nu >0$, and let ${\mathcal{T}}_h$ be a conforming shape-regular affine simplicial mesh of $D$. Let $0=t_0<t_1<\cdots <t_N=T$ be a finite time partition with ${\tau }_n=t_n-t_{n-1}>0$, and put $h={\max }_{K\in {\mathcal{T}}_h}\mathrm{diam}(K)$ and ${\tau }_{\max }={\max }_n{\tau }_n$.

Write $V=H_0^1(D;{\mathbb{R}}^d)$, $Q=L_0^2(D)=\{q\in L^2(D):{\int }_{D}q=0\}$, and $V_{\sigma }=\{v\in V:\mathrm{div}v=0\}$. Let $S_h^1$ be the continuous piecewise-affine scalar space on ${\mathcal{T}}_h$. For each simplex $K$ with barycentric coordinates ${\lambda }_{K,0},\dots ,{\lambda }_{K,d}$, set $b_K=(d+1{)}^{d+1}{\prod }_{j=0}^d{\lambda }_{K,j}$ on $K$ and extend $b_K$ by zero outside $K$. The conforming MINI spaces are $V_h=[S_h^1\cap H_0^1(D){]}^d\oplus {⨁}_{K\in {\mathcal{T}}_h}{b}_K{\mathbb{R}}^d,Q_h=S_h^1\cap Q,$ and their weak divergence kernel is $Z_h=\{v_h\in V_h:b(v_h,q_h)=0\text{ for every }{q}_h\in Q_h\}$. Let $P_{Z_h}$ denote the $L^2(D{)}^d$-orthogonal projection onto this finite-dimensional subspace.

For $v,w\in V$, set $\epsilon (v)=(\mathrm{\nabla }v+\mathrm{\nabla }{v}^{\mathsf{T}})\mathrm{/}2$, $a_{\nu }(v,w)=2\nu {\int }_D\epsilon (v):\epsilon (w)dx$, and $b(v,q)=-{\int }_{D}q\mathrm{div}vdx$. For $a,v,w\in V$, use the real skew convective form $c(a;v,w)=\frac{1}{2}{\int }_D((a\cdot \mathrm{\nabla })v\cdot w-(a\cdot \mathrm{\nabla })w\cdot v)dx\mathrm{.}$ Thus $c(a;v,w)=-c(a;w,v)$ and $c(a;v,v)=0$.

A smooth MINI interface consists of a forcing $f\in L^2(0,T;L^2(D{)}^d)$, an exact no-slip velocity-pressure pair $(u,p)$, and a mixed backward Euler MINI solution $(u_h^n,p_h^n{)}_{n=0}^N$ with the following conditions. The exact pair has $u\in W^{1,\mathrm{\infty }}(0,T;H^2(D{)}^d\cap V_{\sigma })$, $p\in W^{1,\mathrm{\infty }}(0,T;H^1(D)\cap Q)$, and satisfies $({\mathrm{\partial }}_{t}u(t),v{)}_{L^2}+a_{\nu }(u(t),v)+c(u(t);u(t),v)+b(v,p(t))=(f(t),v{)}_{L^2},b(u(t),q)=0$ for every $v\in V$, $q\in Q$, and almost every $t\in (0,T)$. The continuous representatives of $u$ and $p$ are used at the time nodes, and $u_0=u(0)$. The discrete initial value is $u_h^0=P_{Z_h}{u}_0$. With $f_h^n={\tau }_n^{-1}{\int }_{t_{n-1}}^{t_n}f(t)dt$, each $u_h^n\in V_h$ and $p_h^n\in Q_h$ satisfies ${(\frac{u_h^n-u_h^{n-1}}{{\tau }_n},v_h)}_{L^2}+a_{\nu }(u_h^n,v_h)+c(u_h^n;u_h^n,v_h)+b(v_h,p_h^n)=(f_h^n,v_h{)}_{L^2},b(u_h^n,q_h)=0$ for every $v_h\in V_h$, $q_h\in Q_h$, and $1\le n\le N$.

For a family of bounded real-linear velocity sensors ${\mathrm{\ell }}_n\in V^{\ast }$, define the endpoint-sampled exact and discrete observations by $\mathcal{O}(\mathrm{\ell };u)={\sum }_{n=1}^N{\tau }_n{\mathrm{\ell }}_n(u(t_n)),{\mathcal{O}}_h(\mathrm{\ell };u_h)={\sum }_{n=1}^N{\tau }_n{\mathrm{\ell }}_n(u_h^n)\mathrm{.}$ Define the accumulated exact and discrete pressures by $P(t)={\int }_0^{t}p(s)ds\in Q$ and $P_h^0=0$, $P_h^m={\sum }_{n=1}^m{\tau }_n{p}_h^n$. Let ${\bar{p}}_h(t)=p_h^n$ on $(t_{n-1},t_n]$. For $\xi \in Q$ and $\eta \in C_c^{\mathrm{\infty }}(0,T)$, define the accumulated-pressure observation $(P(t_m)-P_h^m,\xi {)}_{L^2}$ and the time-tested pressure observation ${({\int }_0^T(p(t)-{\bar{p}}_h(t))\eta (t)dt,\xi )}_{L^2}$. The latter is an integrated pressure functional; it is not a pointwise-in-time pressure value. This definition supplies inputs and observables only and asserts no error rate or solver-existence result.

Stokes velocity and pressure spaces

real skew convective form

conforming P1-bubble/P1 MINI spaces

weak MINI divergence kernel

mixed backward Euler MINI scheme

Axiom of Countable Choice

Guermond, Chapter 54, §54.2

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