Browsing by Author "Linke, Alexander"
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- ItemAn 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.
- ItemAn assessment of discretizations for convection-dominated convection-diffusion equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Augustin, Matthias; Caiazzo, Alfonso; Fiebach, André; Fuhrmann, Jürgen; John, Volker; Linke, Alexander; Umla, RudolfThe performance of several numerical schemes for discretizing convection-dominated convection-diffusion 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 Streamline-Upwind Petrov--Galerkin (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.
- ItemCollision in a cross-shaped domain(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Linke, AlexanderIn the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed.
- ItemConvergence 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, AlexanderWe 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.
- ItemA discontinuous skeletal method for the viscosity-dependent Stokes problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ern, Alexandre; Di Pietro, Daniele; Linke, Alexander; Schieweck, FriedhelmWe devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree k ≥ 0, and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order (k+1). The main ingredient to achieve pressure-independence is the use of a divergencepreserving velocity econstruction operator in the discretization of the body forces. We also prove an L2-pressure estimate of order (k+1) and an L2-velocity estimate of order (k+2), the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two- and three-dimensional numerical results are presented to support the analysis.
- ItemDivergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Lederer, Philip L.; Linke, Alexander; Merdon, Christian; Schöberl, JoachimClassical inf-sup stable mixed finite elements for the incompressible (Navier-)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. However, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor-Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H (div)-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier-Stokes equations confirm that the new pressure-robust Taylor-Hood and mini elements converge with optimal order and outperform significantly the classical versions of those elements when the continuous pressure is comparably large.
- ItemEfficient 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, ZhenRecent research has shown that in some practically relevant situations like multi-physics flows  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
- ItemA gradient-robust well-balanced scheme for the compressible isothermal Stokes problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Akbas, Mine; Gallouët, Thierry; Gaßmann, Almut; Linke, Alexander; Merdon, ChristianA novel notion for constructing a well-balanced scheme --- a gradient-robust scheme --- is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient --- if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straight-forward extension to barotropic situations with nonlinear equations of state is feasible.
- ItemGuaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Linke, Alexander; Merdon, ChristianThis 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.
- ItemImproving 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.
- ItemInverse 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, MehdiThin 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.
- ItemLocking free and gradient robust H(div)-conforming HDG methods for linear elasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Fu, Guosheng; Lehrenfeld, Christoph; Linke, Alexander; Streckenbach, TimoRobust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.
- ItemMAC schemes on triangular Delaunay meshes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Eymard, Robert; Fuhrmann, Jürgen; Linke, AlexanderWe study two classical generalized MAC schemes on unstructured triangular Delaunay meshes for the incompressible Stokes and Navier-Stokes equations and prove their convergence for the first time. These generalizations use the duality between Voronoi and triangles of Delaunay meshes, in order to construct two staggered discretization schemes. Both schemes are especially interesting, since compatible finite volume discretizations for coupled convection-diffusion equations can be constructed which preserve discrete maximum principles. In the first scheme, called tangential velocity scheme, 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 scheme, called normal velocity scheme, the pressures are defined in the triangles, and the discrete velocities are normal to the edges of the triangles. For both schemes, we prove the convergence in $L^2$ for the velocities and the discrete rotations of the velocities for the Stokes and the Navier-Stokes problem. Further, for the normal velocity scheme, we also prove the strong convergence of the pressure in $L^2$. Linear and nonlinear numerical examples illustrate the theoretical predictions.
- ItemA model of an electrochemical flow cell with porous layer(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Fuhrmann, Jürgen; Linke, AlexanderIn this paper we discuss three different mathematical models for fluid-porous interfaces in a simple channel geometry that appears e.g. in thin-layer channel flow cells. Here the difficulties arise from the possibly different orders of the corresponding differential operators in the different domains. A finite volume discretization of this model allows to calculate the limiting current of the H_2 oxidation in a porous electrode with platinum catalyst particles.
- ItemNew 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
- ItemNon-nested multi-grid solvers for mixed divergence-free Scott-Vogelius discretizations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Linke, Alexander; Matthies, Gunar; Tobiska, LutzStudying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.
- ItemA 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, ChristianMost 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.
- ItemA numerical method for mass conservative coupling between fluid flow and solute transport(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Fuhrmann, Jürgen; Langmach, Hartmut; Linke, AlexanderWe present a new coupled discretization approach for species transport in an incompressible fluid. The Navier-Stokes equations for the flow are discretized by the divergence-free 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 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.
- ItemOn a reduced sparsity stabilization of grad-div type for incompressible flow problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Linke, Alexander; Rebholz, LeoWe 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.
- ItemOn 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, ChristianInf-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.