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Author: Rebholz, Leo G. ×

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Now showing 1 - 9 of 9
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    Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Galvin, Keith J.; Linke, Alexander; Rebholz, Leo G.; Wilson, Nicholas E.
    We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott-Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.
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    An analogue of grad-div stabilization in nonconforming methods for incompressible flows
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Akbas, Mine; Linke, Alexander; Rebholz, Leo G.; Schroeder, Philipp W.
    Grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spacial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization is presented for nonconforming flow discretizations of Discontinuous Galerkin or nonconforming finite element type. Here the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Furthermore, we characterize the limit for arbitrarily large penalization parameters, which shows that the proposed nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit. Several numerical examples illustrate the theory and show their relevance for the simulation of practical, nontrivial flows.
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    On the divergence constraint in mixed finite element methods for incompressible flows
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) John, Volker; Linke, Alexander; Merdon, Christian; Neilan, Michael; Rebholz, Leo G.
    The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which in fluences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations.
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    On the parameter choice in grad-div stabilization for incompressible flow problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Jenkins, Eleanor W.; John, Volker; Linke, Alexander; Rebholz, Leo G.
    Grad-div stabilization has been proved to be a very useful tool in discretizations of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be O(1). This paper revisits this choice for the Stokes equations on the basis of minimizing the H1( ) error of the velocity and the L2( ) error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. Depending on the situation, the optimal stabilization parameter might range from being very small to very large. The analytic results are supported by numerical examples.
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    Improving efficiency of coupled schemes for Navier-Stokes equations by a connection to grad-div stabilitzed projection methods
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Linke, Alexander; Neilan, Michael; Rebholz, Leo G.; Wilson, Nicholas E.
    We prove that in finite element settings where the divergence-free subspace of the velocity space has optimal approximation properties, the solution of Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter , converge to the associated coupled method solution with rate gamma as gamma -> ?. We prove this first for backward Euler schemes, and then extend the results to BDF2 schemes, and finally to schemes with outflow boundary conditions. Several numerical experiments are given which verify the convergence rate, and show how using projection methods in this setting with large grad-div stabilization parameters can dramatically improve accuracy.
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    On the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Linke, Alexander; Rebholz, Leo G.; Wilson, Nicholas E.
    It was recently proven that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be gamma^-frac12 (where gamma is the stabilization parameter), the computational results suggest the rate may be improvable gamma^-1. We prove herein the analytical rate is indeed gamma^-1, and extend the result to other incompressible flow problems including Leray-alpha and MHD. Numerical results are given that verify the theory.
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    New connections between finite element formulations of the Navier-Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bowers, Abigail L.; Cousins, Benjamin R.; Linke, Alexander; Rebholz, Leo G.
    We show the velocity solutions to the convective, skew-symmetric, and rotational Galerkin finite element formulations of the Navier-Stokes equations are identical if Scott-Vogelius elements are used, and thus all three formulations will the same pointwise divergence free solution velocity. A connection is then established between the formulations for grad-div stabilized Taylor-Hood elements: under mild restrictions, the formulations' velocity solutions converge to each other (and to the Scott-Vogelius solution) as the stabilization parameter tends to infinity. Thus the benefits of using Scott-Vogelius elements can be obtained with the less expensive Taylor-Hood elements, and moreover the benefits of all the formulations can be retained if the rotational formulation is used. Numerical examples are provided that confirm the theory
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    Efficient linear solvers for incompressible flow simulations using Scott-Vogelius finite elements
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Cousins, Benjamin; Le Borne, Sabine; Linke, Alexander; Rebholz, Leo G.; Wang, Zhen
    Recent research has shown that in some practically relevant situations like multi-physics flows [11] divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. In order to judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((Pk)d; Pdisc k-1 )) Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements like Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes, and discuss extensions. For iterative methods, we test augmented Lagrangian and H
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    Pressure-induced locking in mixed methods for time-dependent (Navier)Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Linke, Alexander; Rebholz, Leo G.
    We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and NavierStokes 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.