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End date: 2022 × Start date: 2022 × End date: 2024 × Start date: 2020 × Type: report × Version: publishedVersion × Subject: pressure-robustness × WGL Institute: WIAS ×

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Now showing 1 - 5 of 5
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    Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Frerichs, Derk; Merdon, Christian
    Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.
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    Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Lederer, Philip Lukas; Merdon, Christian
    This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1.
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    Pressure-robustness in the context of optimal control
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Merdon, Christian; Wollner, Winnifried
    This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L2-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.
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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.
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    A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Apel, Thomas; Kempf, Volker; Linke, Alexander; Merdon, Christian
    Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case.