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Author: Linke, Alexander × End date: 2015 × Start date: 2013 × DDC: 510 × Has files: Yes ×

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Now showing 1 - 10 of 12
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    On spurious oscillations due to irrotational forces in the Navier-Stokes momentum balance
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Linke, Alexander; Merdon, Christian
    This 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.
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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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    Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Fiebach, André; Glitzky, Annegret; Linke, Alexander
    We investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.
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    Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Fuhrmann, Jürgen; Linke, Alexander; Merdon, Christian; Neumann, Felix; Streckenbach, Timo; Baltruschat, Helmut; Khodayari, Mehdi
    Thin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergence-free finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergence-free velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known real-istic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools.
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    Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Linke, Alexander; Matthies, Gunar; Tobiska, Lutz
    Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.
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    Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Linke, Alexander; Merdon, Christian
    This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix-Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.
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    Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix-Raviart Stokes element
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Linke, Alexander; Merdon, Christian; Wollner, Winnifried
    Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix-Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the energy error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numer-ical experiments in 2D and 3D illustrate the theoretical findings.
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    On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Linke, Alexander
    According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. In order to cure this, a new variational crime using divergence-free velocity reconstructions is proposed. Applying lowest order Raviart-Thomas velocity reconstructions to the nonconforming Crouzeix-Raviart element allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.
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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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    On a reduced sparsity stabilization of grad-div type for incompressible flow problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Linke, Alexander; Rebholz, Leo
    We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operators matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.