Collision in a cross-shaped domain

In 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

An assessment of discretizations for convection-dominated convection-diffusion equations

The 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

A numerical method for mass conservative coupling between fluid flow and solute transport

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 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

Higher order continuous Galerkin--Petrov time stepping schemes for transient convection-diffusion-reaction equations

We present the analysis for the higher order continuous Galerkin--Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a-priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin--Petrov and discontinuous Galerkin time discretization schemes will be given

A numerical method for mass conservative coupling between fluid flow and solute transport

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 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

Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations

This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated

Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations

This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated

Gibbs Phenomena for LqL^q-Best Approximation in Finite Element Spaces -- Some Examples

Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as $q$ tends to $1$. The aim here is to show the potential of $L^1$ as a solution space in connection with suitably designed meshes

Gibbs phenomena for Lq-best approximation in finite element spaces

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as L q-type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for L q-best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over-and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon

Numerical Algorithms for Algebraic Stabilizations of Scalar Convection-Dominated Problems

In dieser Arbeit wurden Finite-Elemente-Verfahren mit algebraischer Fluss\-kor\-rek\-tur (AFC) f\"ur station\"are Konvektions-Diffusions-Reaktions Gleichungen untersucht. Die beiden Hauptaspekte, die studiert wurden, sind iterative L\"oser f\"ur die auftretenden nichtlinearen Gleichungen und adaptive Gitterverfeinerung basierend auf a posteriori Fehlersch\"atzern. Die wichtigsten Ergebnisse der Arbeit sind im Folgenden zusammengefasst. Zun\"achst wurden Studien zu den L\"osern vorgestellt. Es wurden mehrere iterative L\"oser untersucht, darunter Fixpunktans\"atze und Methoden vom Newton-Typ. Die Newton Methoden reduzierten die Anzahl der Iterationen f\"ur bestimmte Beispiele, aber sie waren ineffizient bez\"uglich der Rechenzeit. Der einfachste Fixpunktansatz, n\"amlich \fpr, war auf Grund seiner Matrixeigenschaften am effizientesten. Algorithmische Komponenten, wie die Anderson-Beschleunigung, reduzierten die Anzahl der Iterationen in einigen Beispielen, aber sie lieferte keine Ergebnisse f\"ur den BJK-Limiter. In drei Dimensionen wurde ein iterativer L\"oser f\"ur feinere Gitter ben\"otigt, aber auch hier war \fpr die effizienteste Herangehensweise. Unabh\"angig von der Dimension war es einfacher, die Probleme mit dem Kuzmin-Limiter als mit dem BJK-Limiter zu l\"osen. Der zweite Hauptaspekt sind Studien zur a posteriori Fehlersch\"atzung. Es wurden zwei Ans\"atze zur Bestimmung einer oberen Schranke in der Energie\-norm untersucht, ein auf Resi\-duen basierender Ansatz (\emph{AFC-Energie} Technik) und ein anderer mit der SUPG-L\"osung (\emph{AFC-SUPG-Energie} Technik). Beide Techniken liefern keine robusten Sch\"atzungen bez\"uglich $\varepsilon$, aber es zeigte sich, dass der \emph{AFC-SUPG Energie} Ansatz einen besseren Effektivit\"ats\-index besa{\ss}. F\"ur den BJK-Limiter war die Effektivit\"at besser als f\"ur den Kuzmin-Limiter mit dem \emph{AFC-Energie} Ansatz, w\"ahrend beim \emph{AFC-SUPG Energie} Ansatz die Wahl des Limiters keine Rolle spielte. Im Zuge der adaptiven Gitterverfeinerung kann das Problem lokal diffusions-dominant werden. In diesem Falle muss man den BJK-Limiter verwenden, da man beim Kuzmin-Limiter eine reduzierte Konvergenzordnung beobachten kann. Im Hinblick auf die adaptive Gitterverfeinerung wurden Grenzschichten unterschiedlichen Typs besser mit dem \emph{AFC-Energie} Ansatz verfeinert als mit dem \emph{AFC-SUPG Energie} Ansatz. Schlie{\ss}lich wurden die Ergebnisse f\"ur die a posteriori Fehlersch{\"a}tzung auf Gitter mit h{\"a}ngenden Knoten angewandt. Zun\"achst wurden Ergebnisse bez\"uglich h\"angender Knoten von Lagrange-Elementen niedriger Ordnung auf Elemente h\"oherer Ordnung erweitert. Es zeigte sich in numerischen Studien, dass der Kuzmin-Limiter auf Gittern mit h{\"a}ngenden Knoten dem DMP nicht gen\"ugt, w{\"a}hrend der BJK-Limiter Ergebnisse lieferte, die dem DMP entsprachen. Die Grenzschichten wurden auf konform abgeschlossenen Gittern wesentlich besser approximiert als auf Gittern mit h{\"a}ngenden Knoten. Insgesamt sollte man Gitter mit h{\"a}ngenden Knoten nicht f\"ur AFC Verfahren verwenden.This thesis studies the Algebraic Flux Correction (AFC) schemes for the steady-state convection-diffusion-reaction equations. The work is done on two major aspects of these schemes, namely the iterative solvers for the nonlinear equations and a posteriori error estimation. The major findings of the thesis are summarized below. First, studies concerning the solvers are presented. Several iterative solvers are studied including fixed-point approaches and Newton-type methods. Newton methods reduce the number of iterations for certain examples but it is computationally inefficient. The most simple fixed point approach, namely the fixed point right-hand side is the most efficient because of its matrix structure. Algorithmic components such as Anderson acceleration reduced the number of iterations in some examples but it failed to give results for the BJK limiter. In three dimensions, an iterative solver is needed for finer meshes but here also the fixed point right-hand side is the most efficient. Irrespective of the dimension, it is easier to solve the problem with the Kuzmin limiter as that of the BJK limiter. In conclusion, one might get fewer iterations, with advanced methods but the simple fixed-point approach with dynamic damping is the most efficient in both dimensions. Second, studies for a posteriori error estimation is presented. Two approaches for finding the upper bound are investigated in the energy norm, one residual-based (AFC-Energy technique), and others using the SUPG solution (AFC-SUPG Energy technique). The AFC-Energy estimator is shown not to be robust with respect to $\varepsilon$ and hence, the AFC-SUPG approach gave a better effectivity index. For the BJK limiter, the effectivity is better than the Kuzmin limiter with the AFC-Energy approach, whereas for the AFC-SUPG approach the choice of limiter did not play a role. With adaptive grid refinement, the problem could become locally diffusion dominated and hence one has to use the BJK limiter as one can observe reduced order of convergence for the Kuzmin limiter. In regards to adaptive grid refinement, the AFC-Energy approach approximated the layer much better as compared to the AFC-SUPG approach. Lastly, the results for a posteriori error estimation are extended to grids with hanging nodes. First, results regarding hanging nodes are extended from lower-order Lagrange elements to higher-order elements. It was shown that the Kuzmin limiter fails to satisfy DMP on grids with hanging nodes, whereas the BJK limiter satisfies the DMP. The layers are properly approximated on conformally closed grids in comparison to grids with hanging nodes. Altogether, one should not use grids with hanging nodes for AFC schemes