Refined stability estimates for mixed problems by exploiting semi norm arguments

TL;DR

This paper develops refined stability estimates for mixed problems by utilizing semi norm arguments, linking mathematical analysis to physical regimes and improving accuracy near these regimes.

Contribution

It introduces the use of semi norms on data functionals to derive sharper stability estimates, inspired by pressure-robust discretizations in fluid dynamics.

Findings

Sharper stability estimates near physical regimes

Connection between semi norms and consistency errors

Enhanced understanding of solution behavior in mixed problems

Abstract

Refined stability estimates are derived for classical mixed problems. The novel emphasis is on the importance of semi norms on data functionals, inspired by recent progress on pressure-robust discretizations for the incompressible Navier--Stokes equations. In fact, kernels of these semi norms are shown to be connected to physical regimes in applications and are related to some well-known consistency errors in classical discretizations of mixed problems. Consequently, significantly sharper stability estimates for solutions close to these physical regimes are obtained.