Well-posedness of the Stokes problem under modified pressure Dirichlet boundary conditions

TL;DR

This paper proves the well-posedness of the Stokes problem with boundary conditions where both velocity and pressure vanish, extending inequalities and providing practical guidelines for numerical methods.

Contribution

It introduces a new well-posedness result for the Stokes problem under modified pressure boundary conditions, extending Nečas' inequality.

Findings

Well-posedness established for the modified boundary conditions.

Pressure estimates depend inversely on the boundary band volume.

Numerical experiments validate theoretical results.

Abstract

This paper shows that the Stokes problem is well-posed when velocity and pressure simultaneously vanish on the domain boundary. This result is achieved by extending Ne\v{c}as' inequality to square-integrable functions that vanish in a small band covering the boundary. It is found that the associated a priori pressure estimate depends inversely on the volume of the band. Numerical experiments confirm these findings. Based on these results, guidelines are provided for applying vanishing pressure boundary conditions in model coupling and domain decomposition methods.