Pressure-induced locking in mixed methods for time-dependent (Navier-)Stokes equations

TL;DR

This paper reveals a locking phenomenon in classical mixed methods for time-dependent Stokes and Navier-Stokes equations under force-dominated conditions, and shows pressure-robust methods avoid this issue, with implications for hyperbolic conservation laws.

Contribution

It identifies a new locking phenomenon in classical mixed methods for time-dependent flows and demonstrates pressure-robust methods do not suffer from this, extending understanding of numerical stability.

Findings

Classical methods exhibit reduced or stalled convergence under certain forces.

Pressure-robust methods maintain convergence even in force-dominated regimes.

The analysis connects locking phenomena to well-balanced schemes in hyperbolic PDEs.

Abstract

We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and Navier--Stokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the $L^2$ convergence order for high order methods, and even a complete stall of the $L^2$ convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.