Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

TL;DR

This paper develops a framework for guaranteed velocity error bounds in pressure-robust Stokes discretisations, enabling reliable error control independent of pressure and demonstrating efficiency through numerical tests.

Contribution

It introduces a relaxed constraint framework for pressure-robust discretisations, allowing for pressure-independent error bounds and efficient local flux equilibration.

Findings

Guaranteed upper bounds are close to actual errors.

Framework applies to a wide class of pressure-robust discretisations.

Numerical examples confirm theoretical efficiency.

Abstract

This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager--Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.