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Author: Ahmed, Naveed × Subject: pressure-robustness ×

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Now showing 1 - 2 of 2
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    On 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, Christian
    Inf-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.
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    A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Ahmed, Naveed; Barrenechea, Gabriel R.; Burman, Erik; Guzmán, Johnny; Linke, Alexander; Merdon, Christian
    Discretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(hk+1/2) error estimate in the L2-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.