On the Stokes problem well-posedness under pressure Dirichlet boundary conditions

The Stokes problem is well-posed when considered on an open and bounded domain $\Omega\subset \mathbb{R}^d$ if (i) Dirichlet conditions are imposed for the velocity solution on the boundary $\partial\Omega$ and (ii) the pressure solution is constrained to have zero average on Ω. We find that it is also feasible to impose Dirichlet boundary conditions for the pressure on the entire boundary $\partial\Omega$ as a replacement for (ii). This is done by instead requiring that the pressure function vanishes over a small boundary band $\Delta \subset \Omega$. We theoretically show that this preserves the problem well-posedness, and provide supporting numerical evidence. Applications include couplings of the Stokes subproblems on subdomains of Ω in finite element simulations.