Flux-corrected transport algorithms preserving the eigenvalue range of symmetric tensor quantities

This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using local extremum diminishing (LED) antidi usive corrections of a low order approximation which is assumed to satisfy the relevant inequality constraints. The application of a limiter to antidi usive element contributions guarantees that the corrected solution remains bounded by the local maxima and minima of the low order predictor. The FCT algorithm to be presented in this paper guarantees the LED property for the largest and smallest eigenvalues of the transported tensor at the low order evolution step. At the antidi usive correction step, this property is preserved by limiting the antidi usive element contributions to all components of the tensor in a synchronized manner. The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable alternatives, limiting techniques based on appropriate approximations are considered. The ability of the new limiters to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems

Private Schools in Germany: Attendance up, but Not Among the Children of Less Educated Parents

The percentage of children attending private school in Germany has increased sharply in recent years. According to data of the German Socio-Economic Panel (SOEP), 7% of all students now attend private school. The SOEP, which contains a range of household data, shows that the children of parents with a university entry degree ("Abitur") are more likely to attend private school than those with less educated parents. This trend has become more pronounced in recent years: between 1997 and 2007, the percentage of students with better-educated parents attending private school increased by 77%. By contrast, the corresponding increase for students with less-educated parents was only 12%. Multivariate analyses demonstrate that increasing selection in favor of better-educated groups is particularly evident at the secondary school level. At the primary school level, full-time employment of the mother and regional factors significantly increase the chances of private school attendance. Educational policy should focus on preventing children from better-educated groups from leaving the public school system. If competition among schools is to be encouraged as a matter of policy, efforts should also be made to ensure less educated families consider sending their children to private schools.Private schools, Selection

Combined impact of shifts in Southern Ocean westerlies and Antarctic sea ice during LGM on atmospheric CO2

A significant influence of changes in the westerly winds over the Southern Ocean was proposed as a mechanism to explain a large portion of the glacial atmospheric pCO2 drawdown (Toggweiler et al., 2006). However, additional modelling studies with Earth System Models of Intermediate Complexity do not confirm the size and sometimes even the sign of the impact of southern hemispheric winds on the glacial pCO2 as suggested by Toggweiler (Men- viel et al., 2008; Tschumi et al., 2008, d’Orgeville et al., 2010). We here add to this discussion and explore the potential contribution of changes in the latitudinal position of the winds on Southern Ocean physics and the carbon cycle by using a state-of-the-art ocean general circulation model (MITgcm) in a spatial resolution increasing in the Southern Ocean (2◦ longitude; northern hemisphere: 2◦ latitude; southern hemisphere: 2◦cos(α)). We discuss how the change in carbon cycling is related to the upwelling strength and pattern in the Southern Ocean and how they depend on the changing wind fields and/or the sea ice coverage. While the previous studies explored the impact of the westlies starting from present day or pre-industrial back- ground conditions, we here perform simulations from LGM background climate. Ocean surface conditions are for reasons of consistency taken from output of the COSMOS Earth System model for a pre-industrial control and two LGM runs (Zhang et al., in preparation). Additionally, a northwards shift (by 10◦) of the westerly wind belt as proposed by Toggweiler is investigated

On the design of global-in-time Newton-Multigrid-Pressure Schur complement solvers for incompressible flow problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space-time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES~method and then embedded as a smoother into a space-time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier-Stokes equations by using Newton's method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection-diffusion-reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases

Augmented Lagrangian acceleration of global-in-time Pressure Schur complement solvers for incompressible Oseen equations

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier-Stokes equations

Improving Convergence of Time-Simultaneous Multigrid Methods for Convection-Dominated Problems using VMS Stabilization Techniques

We present the application of a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation for stabilized convection-diffusion equations in the regime of small diffusion coefficients. We use Galerkin finite elements and the Crank-Nicolson scheme for discretization in space and time. The multigrid method blocks all time steps for each spatial unknown, enhancing parallelization in space. While the number of iterations of the solver is bounded above for the 1D heat equation, convergence issues arise in convection-dominated cases. In singularly perturbed advection-diffusion scenarios, Galerkin FE discretizations are known to show instabilities in the numerical solution.We explore a higher-order variational multiscale stabilization, aiming to enhance solution smoothness and improve convergence without compromising accuracy

Fourier analysis of a time-simultaneous two-grid algorithm for the one-dimensional heat equation

In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite di˙erence approximation in space while the semi-discrete problem is discretized in time using the θ-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown. It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence

Numerical Analysis of a Time-Simultaneous Multigrid Solver for Stabilized Convection-Dominated Transport Problems in 1D

The work to be presented focuses on the convection-diffusion equation, especially in the regime of small diffusion coefficients, which is solved using a time-simultaneous multigrid algorithm closely related to multigrid waveform relaxation. For spatial discretization we use linear finite elements, while the time integrator is given by e.g. the Crank-Nicolson scheme. Blocking all time steps into a global linear system of equations and rearranging the degrees of freedom leads to a space-only problem with vector-valued unknowns for each spatial node. Then, common iterative solution techniques, such as the GMRES method with block Jacobi preconditioning, can be used for the numerical solution of the (spatial) problem and allow a higher degree of parallelization in space. We consider a time-simultaneous multigrid algorithm, which exploits space-only coarsening and the solution techniques mentioned above for smoothing purposes. By treating more time steps simultaneously, the dimension of the system of equations increases significantly and, hence, results in a larger number of degrees of freedom per spatial unknown. This can be used to employ parallel processes more efficiently. In numerical studies, the iterative multigrid solution of a problem with up to thousands of blocked time steps is analyzed in 1D. For the special case of the heat equation, it is well known that the number of iterations is bounded above independently of the number of blocked time steps, the time step size, and the spatial resolution. Unfortunately, convergence issues arise for the multigrid solver in convection-dominated regimes. In the context of the standard Galerkin method if the diffusion coefficient is small compared to the grid size and the magnitude of the velocity field, stabilization techniques are typically used to remove artificial oscillations in the solution. However, in our setting, special higher-order variational multiscale-type stabilization methods are discussed, which simultaneously improve the convergence behavior of the iterative solver as well as the smoothness of the numerical solution without significantly perturbing the accuracy

Der Trend zur Privatschule geht an bildungsfernen Eltern vorbei

Der Anteil von Kindern, die eine Privatschule besuchen, hat in den letzten Jahren stark zugenommen. Nach Daten des Sozio-oekonomischen Panels (SOEP) besuchen sieben Prozent aller Schüler eine Privatschule. Die Daten geben auch Auskunft über die Eltern von Privatschülern. Demnach besuchen Schüler, deren Eltern ein Abitur haben, häufiger eine Privatschule als Schüler aus bildungsferneren Elternhäusern. Dieser Trend hat sich in den letzten Jahren verstärkt: So ist der Anteil der Privatschüler aus "bildungsnahen" Elternhäusern zwischen 1997 und 2007 um 77 Prozent gestiegen. Bei Schülern aus bildungsfernen Elternhäusern beträgt der Anstieg lediglich zwölf Prozent. Multivariate Analysen zeigen, dass die zunehmende Selektion hin zu bildungsnahen Schichten insbesondere in der Sekundarstufe auftritt. Im Grundschulbereich erhöhen regionale Faktoren sowie eine Vollerwerbstätigkeit der Mutter die Wahrscheinlichkeit eines Privatschulbesuchs signifikant. Bildungspolitisch sollte verhindert werden, dass insbesondere Kinder aus bildungsnahen Schichten den öffentlichen Schulbereich verlassen. Wird ein Schulwettbewerb grundsätzlich für sinnvoll gehalten sollten Bemühungen unternommen werden, dass auch bildungsferne Schichten Privatschulen in Erwägung ziehen

Private Schools in Germany: Attendance up, but Not Among the Children of Less Educated Parents

The percentage of children attending private school in Germany has increased sharply in recent years. According to data of the German Socio-Economic Panel (SOEP), 7% of all students now attend private school. The SOEP, which contains a range of household data, shows that the children of parents with a university entry degree ("Abitur") are more likely to attend private school than those with less educated parents. This trend has become more pronounced in recent years: between 1997 and 2007, the percentage of students with better-educated parents attending private school increased by 77%. By contrast, the corresponding increase for students with less-educated parents was only 12%. Multivariate analyses demonstrate that increasing selection in favor of better-educated groups is particularly evident at the secondary school level. At the primary school level, full-time employment of the mother and regional factors significantly increase the chances of private school attendance. Educational policy should focus on preventing children from better-educated groups from leaving the public school system. If competition among schools is to be encouraged as a matter of policy, efforts should also be made to ensure less educated families consider sending their children to private schools