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

TL;DR

This paper establishes a domain-independent inf-sup condition for the Stokes problem in slender domains and uses it to develop robust preconditioners, with numerical validation demonstrating effectiveness.

Contribution

It introduces a new pressure norm enabling a stable inf-sup condition independent of domain aspect ratio and applies this to create robust preconditioners.

Findings

Inf-sup constant remains stable regardless of aspect ratio.

Preconditioners show improved robustness in numerical tests.

Theoretical framework extends to slender domain 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.