Publications (32)39.47 Total impact
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ABSTRACT: This paper provides guaranteed upper energy error bounds for a modified lowestorder nonconforming Crouzeix–Raviart finite element method for the Stokes equations. The modification from Linke (Comput Methods Appl Mech Eng 268:782–800, 2014) is based on the observation that only the divergencefree part of the righthand 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 pressureindependent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergencefree part of the righthand 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.Journal of Scientific Computing 08/2015; 64(2):541558. DOI:10.1007/s1091501499439 · 1.70 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 activities of the involved species as primary variables. The numerical scheme works with 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, meshindependent global upper, and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. To illustrate the preservation of qualitative properties by the numerical scheme, we present a longterm simulation of the Michaelis–Menten–Henri system. Especially, we investigate the decay properties of the relative free energy over several magnitudes of time, and obtain experimental orders of convergence for this quantity. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015Numerical Methods for Partial Differential Equations 07/2015; DOI:10.1002/num.21990 · 0.86 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Nearly all infsup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressuredependent, while divergencefree mixed finite elements deliver pressureindependent estimates. A recently introduced new variational crime using lowestorder RaviartThomas velocity reconstructions delivers a much more robust modified CrouzeixRaviart element, obeying an optimal pressureindependent discrete H1 velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowestorder BDM element is proposed, which also allows proving an optimal pressureindependent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the NavierStokes case.  ESAIM Mathematical Modelling and Numerical Analysis 01/2015; DOI:10.1051/m2an/2015044 · 1.64 Impact Factor
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ABSTRACT: We consider discretizations for reactiondiffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, meshindependent global upper and lower L ∞ bounds for the chemical potentials. These bounds provide the main step for a convergence analysis for the full discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete GagliardoNirenberg inequalities. For the proof of the GagliardoNirenberg inequalities we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides with the gradient norm of continuous piecewise linear finite elements.Numerische Mathematik 09/2014; 128(1). DOI:10.1007/s0021101406046 · 1.61 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study the convergence of two generalized markerandcell covolume schemes for the incompressible Stokes and Navier–Stokes equations introduced by Cavendish, Hall, Nicolaides, and Porsching. The schemes are defined on unstructured triangular Delaunay meshes and exploit the Delaunay–Voronoi duality. The study is motivated by the fact that the related discrete incompressibility condition allows to obtain a discrete maximum principle for the finite volume solution of an advection–diffusion problem coupled to the flow.The convergence theory uses discrete functional analysis and compactness arguments based on recent results for finite volume discretizations for the biharmonic equation. For both schemes, we prove the strong convergence in L2 for the velocities and the discrete rotations of the velocities for the Stokes and the Navier–Stokes problem. Further, for one of the schemes, we also prove the strong convergence of the pressure in L2.These predictions are confirmed by numerical examples presented in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014Numerical Methods for Partial Differential Equations 07/2014; 30(4). DOI:10.1002/num.21875 · 0.86 Impact Factor  Advances in Computational Mathematics 04/2014; 40(2). DOI:10.1007/s1044401393161 · 1.49 Impact Factor
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ABSTRACT: 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 wellknown numerical instability of poor mass conservation. The origin of this problem is the lack of L2L2orthogonality between discretely divergencefree velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix–Raviart element is proposed, where divergencefree, lowestorder Raviart–Thomas velocity reconstructions reestablish L2L2orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergencefree 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.Computer Methods in Applied Mechanics and Engineering 01/2014; 268:782–800. DOI:10.1016/j.cma.2013.10.011 · 2.96 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 wellknown 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. blockfull), the matrices arising from the new operator are blockupper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator’s 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.Computer Methods in Applied Mechanics and Engineering 07/2013; s 261–262:142–153. DOI:10.1016/j.cma.2013.04.005 · 2.96 Impact Factor 
Article: Efficient linear solvers for incompressible flow simulations using ScottVogelius finite elements
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ABSTRACT: Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, to appear) divergencefree mixed finite elements may have a significantly smaller discretization error than standard nondivergencefree mixed finite elements. To judge the overall performance of divergencefree mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((Pk)d,Pk1disc) ScottVogelius 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 ScottVogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as TaylorHood 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 LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013Numerical Methods for Partial Differential Equations 07/2013; 29(4). DOI:10.1002/num.21752 · 0.86 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations should be balanced completely by the pressure gradient. Unfortunately, nearly all available discretizations for incompressible flows violate this property. The origin of the problem is that discrete velocities are usually not divergencefree. Hence, the use of divergencefree velocity reconstructions is proposed wherever an L2L2 scalar product appears in the discrete variational formulation. The approach is illustrated and applied to a nonconforming MAClike discretization for unstructured Delaunay grids. It is numerically demonstrated that a divergencefree velocity reconstruction based on the lowestorder Raviart–Thomas element increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.RésuméLors dʼécoulements incompressibles avec vitesses normales nulles à la frontière, les forces présentes dans les équations de conservation de la quantité de mouvement dont le rotationnel sʼannule ne doivent tre équilibrées que par le gradient de la pression. Malheureusement, cette propriété nʼest pas vérifiée par la plupart des méthodes de discrétisation disponibles, pour lesquelles la divergence (en un sens continu) de lʼapproximation de la vitesse nʼest pas nulle. Aussi, nous proposons dʼutiliser une reconstruction continue de la vitesse à divergence nulle, dans chaque produit scalaire L2L2 intervenant dans la formulation variationnelle. Nous illustrons cette méthode dans le cas dʼun schéma non conforme de type MAC sur grille non structurée de Delaunay. La reconstruction basée sur les éléments de Raviart–Thomas de bas degré, permet dʼaccroître la robustesse et la précision de ce schéma dans des cas de forces irrotationnelles significatives.Comptes Rendus Mathematique 09/2012; 350(1718):837840. DOI:10.1016/j.crma.2012.10.010 · 0.47 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotationfree forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergencefree finite elements (such as Scott–Vogelius), or heavy graddiv stabilization of weakly divergencefree elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.Computer Methods in Applied Mechanics and Engineering 09/2012; s 237–240:166–176. DOI:10.1016/j.cma.2012.05.008 · 2.96 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions are constructed and analyzed, first on grids which satisfy an orthogonality condition, and then on general, possibly non conforming meshes. In both cases, the piecewise constant approximate solution is shown to converge in L2 () to the exact solution; similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. These results are confirmed by numerical results.Mathematics of Computation 01/2012; 81:20192048. DOI:10.1090/S002557182012026081 · 1.49 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The performance of several numerical schemes for discretizing convectiondominated convectiondiffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the StreamlineUpwind PetrovGalerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.Computer Methods in Applied Mechanics and Engineering 11/2011; 200:33953409. DOI:10.1016/j.cma.2011.08.012 · 2.96 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: It was recently proven in Case et al. (2010) [2] that, under mild restrictions, graddiv 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 γ−12 (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to γ−1. We prove herein the analytical rate is indeed γ−1, and extend the result to other incompressible flow problems including Lerayα and MHD. Numerical results are given that verify the theory.Journal of Mathematical Analysis and Applications 09/2011; 381(2):612626. DOI:10.1016/j.jmaa.2011.03.019 · 1.12 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier–Stokes equations for the flow are discretized by the divergencefree Scott–Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection–diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.Applied Numerical Mathematics 04/2011; 61(461):530553. DOI:10.1016/j.apnum.2010.11.015 · 1.22 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier—Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott—Vogelius element pair has recently been shown to be infsup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier—Stokes equations. We also prove that the limit of the graddiv stabilized Taylor—Hood solutions to the Navier—Stokes problem converges to the Scott—Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the graddiv parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott—Vogelius and graddiv stabilized Taylor—Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier—Stokes approximations.SIAM Journal on Numerical Analysis 01/2011; 49(4):14611481. DOI:10.2307/23074341 · 1.79 Impact Factor 
Conference Paper: Mass Conservative Coupling Between Fluid Flow and Solute Transport
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ABSTRACT: We present a coupled discretization approach for species transport in an incompressible fluid. The NavierStokes equations for the flow are discretized by the divergencefree ScottVogelius element. The convectiondiffusion equation for species transport is discretized by the Voronoi finite volume method. The species concentration fulfills discrete global and local maximum principles. We report convergence results for the coupled scheme and an application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell.Finite Volumes for Complex Applications VI; 01/2011  [Show abstract] [Hide abstract]
ABSTRACT: We study extensions of the energy and helicity preserving scheme for the 3D NavierStokes equations, developed in (23), to a more general class of problems. The scheme is studied together with stabilizations of graddiv type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme.International Journal of Numerical Analysis and Modeling 01/2011; 8(1). · 0.82 Impact Factor 
Chapter: MAC Schemes on Triangular Meshes
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ABSTRACT: We present numerical results for two generalized MAC schemes on triangular meshes, which are based on staggered meshes using the Delaunay–Voronoi duality. In the first one, the pressures are defined at the vertices of the mesh, and the discrete velocities are tangential to the edges of the triangles. In the second one, the pressures are defined in the triangles, and the discrete velocities are normal to the edges of the triangles. In both cases, convergence results are obtained. KeywordsNavier–StokesMAC schemeDelaunay mesh12/2010: pages 399407;
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203  Citations  
39.47  Total Impact Points  
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Institutions

20082015

Weierstrass Institute for Applied Analysis and Stochastics
Berlín, Berlin, Germany


20122014

Freie Universität Berlin
 Department of Mathematics and Computer Science
Berlín, Berlin, Germany
