Refined stability estimates for mixed problems by exploiting semi norm arguments

2025/06/13 by Nicolas R. Gauger, Gauger, Nicolas, Alexander Linke +3
Decision Sciences · Engineering · Mathematics · #35B35 #65J05 #65N12 #65N30 #76D05 #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design #Stability and Control of Uncertain Systems

paper · pdf · doi:10.48550/arxiv.2506.11566

openalex publication_date 2025/06/13 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

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.

Citations

• Guaranteed upper bounds for the velocity error of pressure-robust Stokes\n discretisations
• A pressure-robust discretization of Oseen's equation using stabilization\n in the vorticity equation
• Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem
• On high-order pressure-robust space discretisations, their advantages\n for incompressible high Reynolds number generalised Beltrami flows and beyond
• Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods
• Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes

Related

• Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem
• Well-posedness for the Navier-Stokes equations in critical mixed-norm Lebesgue spaces
• Wellposedness and stability results for the Navier-Stokes equations in \mathbf R³
• Mixed-norm L_p-estimates for non-stationary Stokes systems with singular VMO coefficients and applications
• Stability in terms of two measures of semiflows in space conv(Rⁿ)