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

TL;DR

The paper addresses the well-posedness of the Stokes equations when velocity vanishes on the boundary and pressure vanishes on a near-boundary band $\Delta$ by extending Nečas' inequality to $L^2_\Delta(\Omega)$. The authors establish an inf-sup stability and derive a priori bounds for both pressure and velocity, with the pressure bound scaling as $|\Delta|^{1/2}$ in the worst case. They develop a rigorous operator framework using projections $P$ and $T$ to extend the Nečas inequality and corroborate the theory with finite-element simulations using Taylor–Hood elements. The results yield practical guidelines for applying Dirichlet-type pressure boundary conditions in domain decomposition and model-coupling contexts, ensuring well-posedness and controlled pressure growth as the band size varies.

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 Nečas' 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.

Figures (11)

• Figure 1: The flower shaped domain is $\Omega$. Different choices of region $\Delta \subset \Omega$ are shaded using grey color. In the three images from left to right: $\Delta$ is a zero set, $\Delta$ is a small band around the boundary of $\Omega$, and $\Delta$ is a small patch in the interior of $\Omega$.
• Figure 2: Piecewise discontinuous constant pressure functions vanishing on one part of $\partial\Omega$ (left) and vanishing on the shaded area $\Delta \subset \Omega$ close to one part of $\partial\Omega$ (right), where $\Omega \subset \mathbb{R}$ is an open and bounded domain.
• Figure 3: The value of constant $C = \|Tp\|_{L^2(\Omega)}/\|p\|_{L^2(\Omega)}$ as the volume $|\Delta|$ is refined, for three different example pressure functions \ref{['eq:experiments:constant:pressure_functions']}. In plots 1, 3, and 4 (from left to right) $\Delta \subset \Omega$ is a disk centred within a unit disk $\Omega$. In plot 2 (from left to right) $\Delta \subset \Omega$ is a ring around $\partial\Omega$.
• Figure 4: Condition numbers of the stiffness matrix, when the pressure solution function is constrained to: (i) have zero mean on $\Omega$ (label Mean zero), (ii) vanish over $\partial \Omega$ (label Boundary zero), (iii) vanish over $\Delta \subset \Omega$ (label Band $\Delta$ zero). The condition numbers are plotted as a function of volume of $\Delta\subset \Omega$ when mesh size is fixed (right), and as a function of mesh size (left). In the right plot $\Delta$ is an area centred around a point $(1,0)$. In the left plot $\Delta$ is a ring containing triangles closest to domain boundary $\partial\Omega$.
• Figure 5: The domain subdivision used for coupling an exact Stokes problem over $\Omega_{\text{sub}}\subset\Omega$ to a perturbed Stokes problem over $\Omega\backslash\Omega_{\text{sub}}$, where $\Delta\subset\Omega_{\text{sub}}$ is a small band where Dirichlet boundary conditions are imposed. Left: $D(\Delta) = 0, |\Delta|=0$. Right: $D(\Delta) > 0, |\Delta|>0$.

Theorems & Definitions (14)

• theorem 1: Babuška-Brezzi, Theorem 49.3 in guermond_volii
• definition thmcounterdefinition: Operator $P$ and $T$
• lemma thmcounterlemma: Properties of operator $T$
• proof
• lemma thmcounterlemma: Properties of operator $P$
• proof
• lemma thmcounterlemma: Range and kernel of $T$
• proof
• theorem 2: Extended Nečas inequality
• proof