$\textit{A Priori}$ Error Analysis for the $p$-Stokes Equations with Slip Boundary Conditions: A Discrete Leray Projection Framework

TL;DR

The paper develops an a priori error framework for the unsteady $p$-Stokes system with impermeability and Navier slip boundary conditions, focusing on both velocity and kinematic pressure under fully-discrete schemes. A central tool is a discrete Leray projection that converges linearly to its continuous counterpart, enabling a discrete Helmholtz decomposition and leading to (quasi-)best-approximation error estimates. By combining this projection framework with non-Newtonian regularity theory, the authors derive explicit rates for velocity and pressure errors that remain robust under reduced regularity. The results are corroborated by numerical experiments using Taylor–Hood elements in FEniCS, demonstrating the predicted convergence behavior and the stability of the discrete Leray projections in practice.

Abstract

We present an $\textit{a priori}$ error analysis for the kinematic pressure in a fully-discrete finite-differences/-elements discretization of the unsteady $p$-Stokes equations, modelling non-Newtonian fluids. This system is subject to both impermeability and perfect Navier slip boundary conditions, which are incorporated either weakly via Lagrange multipliers or strongly in the discrete velocity space. A central aspect of the $\textit{a priori}$ error analysis is the discrete Leray projection, constructed to quantitatively approximate its continuous counterpart. The discrete Leray projection enables a Helmholtz-type decomposition at the discrete level and plays a key role in deriving error decay rates for the kinematic pressure. We derive (in some cases optimal) error decay rates for both the velocity vector field and kinematic pressure, with the error for the kinematic pressure measured in an $\textit{ad hoc}$ norm informed by the projection framework. The $\textit{a priori}$ error analysis remains robust even under reduced regularity of the velocity vector field and the kinematic pressure, and illustrates how the interplay of boundary conditions and projection stability governs the accuracy of pressure approximations.

Figures (1)

• Figure 1: Computed $c_{\textup{stab}}^{i}(\mathcal{J}_{{h_i}})$, $\mathcal{J}_{{h_i}}\in \{\mathcal{P}_{{h_i}},\mathcal{P}_{{h_i}}^{\perp}\}$, $i=1,\ldots,39$: left:\ref{['eq:bc.1']} is strongly imposed; right:\ref{['eq:bc.1']} is weakly imposed.

Theorems & Definitions (56)

• Lemma 2.1
• proof
• Lemma 2.2: $\varepsilon$-Young inequality
• proof
• Lemma 2.3: shift-change
• proof
• Definition 2.4: weak formulation
• Lemma 2.5: inf-sup stability for $(\widehat{\bfV},Q,\widehat{Z})$
• Remark 2.6
• proof : Proof (of Lemma \ref{['lem:inf-sup_continuous']})