5 The Weak Stokes Problem
5.1 A model mixed problem
For steady incompressible Stokes flow with homogeneous velocity boundary conditions, seek velocity \(\mathbf u\) and pressure \(p\) satisfying
\[-\mu\Delta\mathbf u+\nabla p=\mathbf f,\qquad\nabla\cdot\mathbf u=0.\]
A common variational form uses \(V=[H_0^1(\Omega)]^d\) and \(Q=L_0^2(\Omega)\), where the pressure space has zero mean to remove the constant-pressure ambiguity under the stated boundary conditions.
Find \((\mathbf u,p)\in V\times Q\) such that, for all \((\mathbf v,q)\in V\times Q\),
\[\mu\int_\Omega\nabla\mathbf u:\nabla\mathbf v\,d\Omega-\int_\Omega p\,\nabla\cdot\mathbf v\,d\Omega=\int_\Omega\mathbf f\cdot\mathbf v\,d\Omega,\]
\[\int_\Omega q\,\nabla\cdot\mathbf u\,d\Omega=0.\]
This is a simplified constant-viscosity form. Boundary tractions and alternative symmetric-gradient formulations require corresponding adjustments.
5.2 A saddle-point structure
After discretization, the algebraic system commonly has the block structure
\[\begin{bmatrix}A&B^T\\B&0\end{bmatrix}\begin{bmatrix}\mathbf U\\P\end{bmatrix}=\begin{bmatrix}F\\0\end{bmatrix},\]
with sign conventions depending on the definition of \(B\). The zero block reflects the constraint structure rather than an omitted physical effect.
Think about it: Why must the pressure reference be fixed for many closed or fully velocity-prescribed systems?
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.