Pressure-induced locking in mixed methods for time-dependent (Navier--)Stokes equations
- Linke, Alexander
- Rebholz, Leo G.
2010 Mathematics Subject Classification
- 76D05 65M60 35M13
- Time-dependent Stokes equations, Navier--Stokes equations, mixed finite elements, pressure-robustness, structure-preserving, space discretization, well-balanced schemes
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 L2 convergence order for high order methods, and even a complete stall of the L2 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.
- J. Comput. Phys., 388 (2019), pp. 350--356, DOI 10.1016/j.jcp.2019.03.010 .