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- ItemOn spurious oscillations due to irrotational forces in the Navier-Stokes momentum balance(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Linke, Alexander; Merdon, ChristianThis contribution studies the in uence of the pressure on the velocity error in finite element discretisations of the Navier-Stokes equations. Three simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. For widely used finite element methods like the Taylor-Hood finite element method, such relatively large pressures can lead to spurious oscillations and arbitrarily large errors in the velocity, even if the exact velocity is in the ansatz space. Only mixed finite element methods, whose velocity error is pressure-independent, like the Scott-Vogelius finite element method can avoid this influence.
- ItemOn really locking-free mixed finite element methods for the transient incompressible Stokes equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Ahmed, Naveed; Linke, Alexander; Merdon, ChristianInf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities.
- ItemQuasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Linke, Alexander; Merdon, Christian; Neilan, Michael; Neumann, FelixNearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressurerobust scheme with low regularity. The numerical analysis applies divergence-free H1-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming CrouzeixRaviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.
- ItemPressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Linke, Alexander; Merdon, ChristianRecently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new pressure-robust Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, pressure-robustness is extended to the time-dependent Navier-Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the discrete Helmholtz projector of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed.