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

TL;DR

This work develops a unified, abstract framework for hybrid velocity-pressure discretizations of the Stokes problem, establishing stability and error estimates that separate velocity and pressure contributions. By constructing local reconstructions for the velocity divergence and pressure gradient, and combining them with a velocity gradient reconstruction and stabilization, the authors prove (generalized) inf-sup stability and derive velocity- and pressure-error decompositions. The framework is then instantiated for several schemes, including the classical Botti--Massa method, the Rhebergen--Wells method, and two new RTN- and polytopal-based approaches, with rigorous analysis and extensive numerical validation demonstrating optimal convergence and pressure-robustness under appropriate inclusions and stabilization. Significantly, when the discrete spaces satisfy the key inclusion properties and the global IBP condition, velocity errors can be made independent of the pressure (and viscosity), ensuring robust performance in quasi-inviscid regimes and enabling accurate velocity fields on complex polyhedral meshes.

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.

Figures (4)

• Figure 1: "tri"
• Figure 2: "cart"
• Figure 3: "locref"
• Figure 4: "hexa"

Theorems & Definitions (27)

• Proposition 1: Commutativity of $D_T$
• proof
• Proposition 2: Commutativity of $G_T$
• Lemma 3: Inf-sup and generalized inf-sup conditions
• proof
• Proposition 4: Commutativity of $E_T$
• Lemma 5: Stability of the scheme
• Remark 6: A priori estimate
• proof : Proof of Lemma \ref{['lem:stability']}
• Lemma 7: Error estimate