The incompressible Stokes problem for the vector field u and the scalar pressure field p is governed by the following equations:
$$ - \operatorname{div} \left({v\user2{\varepsilon }(\user2{u})} \right) + \nabla p = \user2{b}\quad \quad \operatorname{in} \;\Omega ,$$
(8.1)
$$\operatorname{div} \;\user2{u} = 0\quad \quad \operatorname{in} \;\Omega ,$$
(8.2)
$$\user2{u} = {\user2{g}^D}\quad \quad \operatorname{on} \;{\Gamma ^D},$$
(8.3)
$$v\user2{\varepsilon }\left(\user2{u} \right) = {\user2{g}^N}\quad \quad \operatorname{on} \;{\Gamma ^N},$$
(8.4)
where v > 0 is the fluid viscosity, the vector-valued field b is the forcing term, the vector-valued fields g D and g N are the boundary data, and e(u) = (∇u + (∇u) T )/2 is the symmetric strain tensor. We refer the reader to Sect. 1.5.1 for a more detailed presentation of the Stokes problem.