Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain $\Omega \subset \mathbb{R}^3$. We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity field, pressure, and magnetic fields at every iteration step

Optimal control for the thin-film equation: Convergence of a multi-parameter approach to track state constraints avoiding degeneracies

We consider an optimal control problem subject to the thin-film equation which is deduced from the Navier--Stokes equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness, and the rigorous derivation of necessary optimality conditions for the optimal control problem is performed. A multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint, and convergence is shown for vanishing regularization parameters by decoupling both effects. The fully regularized optimal control problem allows for practical simulations which are provided, including the control of a dewetting scenario, to evidence the need of the state constraint, and to motivate proper scalings of involved regularization and numerical parameters

Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise

The stochastic Allen-Cahn equation with multiplicative noise involves the nonlinear drift operator ${\mathscr A}(x) = \Delta x - \bigl(\vert x\vert^2 -1\bigr)x$. We use the fact that ${\mathscr A}(x) = -{\mathcal J}^{\prime}(x)$ satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate $\underset{1 \leq j \leq J}\sup {\mathbb E}\bigl[ \Vert X_{t_j} - Y^j\Vert_{{\mathbb L}^2}^2\bigr] \leq C_{\delta}(k^{1-\delta} + h^2)$ for all small $\delta>0$, where $X$ is the strong variational solution of the stochastic Allen-Cahn equation, while $\big\{Y^j:0\le j\le J\big\}$ solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh $\{ t_j;\, 1 \leq j \leq J\}$ of size $k>0$ which covers $[0,T]$

Sustainability of Swiss Fiscal Policy

We examine whether Swiss federal fiscal policy was sustainable over the period from 1900 to 2002. We perform unit root and cointegration tests for federal revenues and expenditures, taking into account a structural shift in the budgetary process related to World War II. We find sustainability over the entire period. However, splitting the sample into two sub-samples before and after World War II, the results do much less support sustainability. Finally, applying the tax smoothing model of BARRO (1979), we show that cyclical fluctuations of the output and changes in expected inflation rate are major determinants of the federal budgetdeficit over the time period considered.sustainability, budget deficit, cointegration, structural breaks

Sustainability of Swiss fiscal policy

We examine whether Swiss federal fiscal policy was sustainable over the period from 1900 to 2002. We perform unit root and cointegration tests for federal revenues and expenditures, taking into account a structural shift in the budgetary process related to World War II. We find sustainability over the entire period. However, splitting the sample into two sub-samples before and after World War II, the results do much less support sustainability. Finally, applying the tax smoothing model of BARRO (1979), we show that cyclical fluctuations of the output and changes in expected inflation rate are major determinants of the federal budget deficit over the time period considered

Multiscale resolution in the computation of crystalline microstructure

Summary.: This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optima

Sustainability of Public Debt and Budget Deficit: Panel cointegration analysis for the European Union Member countries

In this study, we analyse the sustainability of fiscal policy of EU member countries within the panel cointegration and error-correction frameworks. Unlike the previous empirical papers in this area, we apply the test for panel cointegration between the primary budget deficit and the public debt defined in GDP ratios. Based on the cointegration test results, we conclude that the fiscal policy is consistent with the intertemporal budget constraint, i.e., it is sustainable in the panel of fifteen EU member countries over the period from 1970 to 2004. Hence, we show that the fiscal balance exhibits a significant structural change in the year 1992, when we apply the Banerjee and Carrion-i-Silvestre (2006) test for a structural break in the panel cointegration relationship. In a next step, we search for the politico-economic factors which explain the variation in the sustainable fiscal balance among the European countries. We evidence that the European fiscal rules have a significant positive effect on the improvement of the fiscal position of the governments of the EU member countries

Stabilization methods in relaxed micromagnetism

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potential u and magnetization m. In [C. Carstensen and A. Prohl, Numer. Math. 90 (2001) 65–99], the conforming P1 - (P0)d-element in d=2,3 spatial dimensions is shown to lead to an ill-posed discrete problem in relaxed micromagnetism, and suboptimal convergence. This observation motivated a non-conforming finite element method which leads to a well-posed discrete problem, with solutions converging at optimal rate. In this work, we provide both an a priori and a posteriori error analysis for two stabilized conforming methods which account for inter-element jumps of the piecewise constant magnetization. Both methods converge at optimal rate; the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ]