Stability, convergence, and pressure-robustness of numerical schemes for incompressible flows with hybrid velocity and pressure

TL;DR

This paper analyzes the stability, convergence, and pressure-robustness of discretization schemes for incompressible flows, focusing on the Stokes problem, and provides theoretical insights along with numerical validation.

Contribution

It introduces a framework for assessing pressure-robustness and stability of hybrid velocity-pressure schemes, including new criteria and examples.

Findings

Identifies assumptions for inf-sup stability.

Derives error estimates separating velocity and pressure errors.

Demonstrates pressure-robustness under specific properties.

Abstract

In this work we study the stability, convergence, and pressure-robustness of discretization methods for incompressible flows with hybrid velocity and pressure. Specifically, focusing on the Stokes problem, we identify a set of assumptions that yield inf-sup stability as well as error estimates which distinguish the velocity- and pressure-related contributions to the error. We additionally identify the key properties under which the pressure-related contributions vanish in the estimate of the velocity, thus leading to pressure-robustness. Several examples of existing and new schemes that fit into the framework are provided, and extensive numerical validation of the theoretical properties is provided.