Pressure-induced locking in mixed methods for time-dependent (Navier-)Stokes equations
For researchers using mixed finite element methods for incompressible flow, this work reveals a previously unrecognized locking effect and demonstrates the advantage of pressure-robust methods.
The paper identifies a pressure-induced locking phenomenon in inf-sup stable mixed methods for time-dependent Stokes and Navier-Stokes equations, showing that classical methods like Taylor-Hood suffer from reduced L^2 convergence order, while pressure-robust methods avoid this issue.
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.