Stokes velocity and pressure spaces and weak forms

Assume Countable Choice. Let $d\in \{2,3\}$ and let $D\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz open set, where Lipschitz means that $\mathrm{\partial }D$ has a finite rigidly moved graph-chart atlas. Here $H_0^1(D)$ is the $H^1$-closure of $C_c^{\mathrm{\infty }}(D)$ (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms). Since $D$ has finite measure, Holder makes every $L^2$ class integrable, so the zero-mean condition is well-defined (Holder's inequality for integrals, including the endpoint cases). Define $V:=H_0^1(D;{\mathbb{R}}^d),Q:=L_0^2(D):=\{q\in L^2(D;\mathbb{R}):{\int }_{D}q=0\}\mathrm{.}$ Equip $V$ with $\mathrm{\parallel }v{\mathrm{\parallel }}_V:=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D)}$ and $Q$ with $\mathrm{\parallel }q{\mathrm{\parallel }}_Q:=\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(D)}$. The local Poincare inequality A bounded-domain Poincare inequality for H1-zero fields makes the first a norm equivalent to the usual $H^1$ norm. For $v\in V$, define $\epsilon (v):=(\mathrm{\nabla }v+\mathrm{\nabla }{v}^T)\mathrm{/}2$ and $b(v,q):=-{\int }_{D}q\mathrm{div}v$. For viscosity $\nu >0$, set $a_{\nu }(u,v):=2\nu {\int }_D\epsilon (u):\epsilon (v)$, where $X:Y={\sum }_{i,j}{X}_{ij}{Y}_{ij}$. The continuous domain class is all bounded connected Lipschitz $D$; the exact mini-element mesh class is the narrower polygonal/polyhedral conforming affine simplicial class.