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

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