On a robust inf-sup condition for the Stokes problem in slender domains -- with application to preconditioning

TL;DR

The paper addresses robust inf-sup stability for the Stokes equations in slender domains by introducing a weakened pressure norm $\|q\|_*$ that is spectrally equivalent to the sum-norm $\|q\|_{L^2+WH^1}$. This reformulation yields an inf-sup constant independent of the domain aspect ratio, enabling domain-robust operator preconditioning. The authors develop two robust preconditioners, $\mathcal{B}$ and $\mathcal{B}_H$, incorporating a coarse-space Laplacian and a dimensionally reduced model to stabilize the pressure space. Theoretical results are complemented by numerical experiments on channel-like geometries and 3D Tesla-valve-inspired domains, showing bounded iteration counts and condition numbers across varying $L/W$ and discretizations. The work advances robust fast solvers for Stokes in slender geometries with implications for simulations in microfluidics and coupled multi-physics problems.

Abstract

We identify a norm on the pressure variable in the Stokes equation that allows us to prove a continuous inf-sup condition with a constant independent of the domain's aspect ratio. This is in contrast to the standard inf-sup constant, which breaks down as the aspect ratio increases. We further apply our result to construct robust operator preconditioners for the Stokes problem in slender domains. Several numerical examples illustrate the theory.

Figures (7)

• Figure 1: Eigenvalue problem for the Stokes problem on channel $L=20$ and mixed boundary conditions, where velocity is set on long boundaries, using preconditioner \ref{['eq:B_standard']}. Shown are the velocity (as glymphs) and pressure components of the eigenmodes corresponding to the first four smallest in magnitude eigenvalues; $\lvert \lambda_0 \rvert \approx 2\cdot10^{-3}$, $\lvert \lambda_1 \rvert \approx 8\cdot10^{-3}$, $\lvert \lambda_2 \rvert \approx 18\cdot10^{-2}$, $\lvert \lambda_3 \rvert \approx 31\cdot10^{-2}$. For the first mode, the inf-sup constant is attained.
• Figure 1: Pressure-driven Stokes flow scenario in a simple channel domain (large aspect ratio, $L/W \ll 1$, small angle) discretized by coarse grid discretization with elements $\Omega_i$ (approximate aspect ratio $1$) and element faces $F_i$.
• Figure 1: Coarse partitioning of domain $\Omega=(0, 10)\times (0, 1)$, cf. \ref{['sec:theory']}, in terms of quadrilateral mesh $\Omega_H$ with $1\times 1$ cells $\Omega_i$, $1\leq i \leq 10$. Operator $-\Delta_H$ in \ref{['eq:precond_manfred']} is discretized on the space $Q_H$ of piecewise constant functions with respect to $\Omega_H$ leading to a "one-dimensional" Laplacian (with its "segment" mesh in red). Pressure space $Q_h$ is defined over the triangular mesh. Here the mesh for refinement level $l=0$, cf. \ref{['ex:noslip']}, is shown.
• Figure 2: Channel with constrictions. (Top) Parameterization of the constriction in terms of parameter $r$. (Bottom) Thickness functions $W$ for case $r=0.4$. With \ref{['eq:precond_espen']} minimal thickness is used, $\min_{\mathbf{x}}W(\mathbf{x})$, while with \ref{['eq:precond_varW']}$W$ varies in space.
• Figure 3: Performance of preconditioners \ref{['eq:precond_espen']} for Stokes problem in the channel domain $(0, 10)\times(0, W)$ with no-slip boundary conditions on top and bottom. Discretization by (left) $[\mathbb{P}_2]^2\times\mathbb{P}_1$ and (right) $[\mathbb{P}_2]^2\times\mathbb{P}_0$ elements. In the subplots, condition numbers and MinRes iterations are plotted on the left and right vertical axes, respectively. The values are obtained on meshes with refinement level $l=4$ ($h=3\cdot10^{-2}$) with the errorbars representing difference to level $l=3$ results.

Theorems & Definitions (26)

• Remark 1
• Remark 2
• Example 1: No-slip and traction boundaries
• Example 2: Free-slip boundary
• Example 3: The univariate case
• Lemma 1
• Proof 1
• Lemma 2
• Proof 2
• Lemma 3