Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00211-016-0808-zFor the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions

A unified analysis of Algebraic Flux Correction schemes for convection-diffusion equations

Recent results on the numerical analysis of Algebraic Flux Correction (AFC) finite element schemes for scalar convection-diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion scheme. Then, specific versions of the method, this is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme

A unified analysis of algebraic flux correction schemes for convection–diffusion equations

Recent results on the numerical analysis of algebraic flux correction (AFC) finite element schemes for scalar convection–diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion schemes. Then, specific versions of the method, that is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme

A numerical assessment of finite element discretizations for convection-diffusion-reaction equations satisfying discrete maximum principles

Numerical studies are presented that investigate finite element methods satisfying discrete maximum principles for convection-diffusion-reaction equations. Two linear methods and several nonlinear schemes, some of them proposed only recently, are included in these studies, which consider a number of two-dimensional examples. The evaluation of the results examines the accuracy of the numerical solutions with respect to quantities of interest, like layer widths, and the efficiency of the simulations

Maximum-principle preserving space-time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions

Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation

This work is devoted to the proposal, analysis, and numerical testing of a fully computable a posteriori error bound for a class of nonlinear discretizations of the convection-diffusion-reaction equation. The type of discretization we consider is nonlinear, since it has been built with the aim of preserving the discrete maximum principle. Under mild assumptions on the stabilizing term, we obtain an a posteriori error estimator that provides a certified upper bound on the norm of the error. Under the additional assumption that the stabilizing term is both Lipschitz continuous and linearity preserving, the estimator is shown to be locally efficient. We present examples of discretizations that satisfy these two requirements, and the theory is illustrated by several numerical experiments in two and three space dimensions

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

Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws

We present a new perspective on the use of weighted essentially nonoscillatory (WENO) reconstructions in high-order methods for scalar hyperbolic conservation laws. The main focus of this work is on nonlinear stabilization of continuous Galerkin (CG) approximations. The proposed methodology also provides an interesting alternative to WENO-based limiters for discontinuous Galerkin (DG) methods. Unlike Runge--Kutta DG schemes that overwrite finite element solutions with WENO reconstructions, our approach uses a reconstruction-based smoothness sensor to blend the numerical viscosity operators of high- and low-order stabilization terms. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while preserving the high-order accuracy of a linearly stable baseline discretization in regions where the exact solution is sufficiently smooth. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive error estimates and perform numerical experiments. In particular, we prove that the consistency error of the nonlinear stabilization is of the order $p+1/2$, where $p$ is the polynomial degree. This estimate is optimal for general meshes. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as $p+1$