A PDAE formulation of parabolic problems with dynamic boundary conditions

The weak formulation of parabolic problems with dynamic boundary conditions is rewritten in form of a partial differential-algebraic equation. More precisely, we consider two dynamic equations with a coupling condition on the boundary. This constraint is included explicitly as an additional equation and incorporated with the help of a Lagrange multiplier. Well-posedness of the formulation is shown

A multiscale method for heterogeneous bulk-surface coupling

In this paper, we construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As origin, we consider a reformulation of the system in order to decouple the discretization of bulk and surface dynamics. This allows us to combine multiscale methods on the boundary with standard Lagrangian schemes in the interior. We prove convergence and quantify explicit rates for low-regularity solutions, independent of the oscillatory behavior of the heterogeneities. As a result, coarse discretization parameters, which do not resolve the fine scales, can be considered. The theoretical findings are justified by a number of numerical experiments including dynamic boundary conditions with random diffusion coefficients

Splitting methods for constrained diffusion-reaction systems

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ODE. However, Strang splitting suffers from order reduction which limits its efficiency. This is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost. Numerical examples including a coupled mechanical system illustrate the proven convergence results

Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials

This paper concerns spectral properties of linear Schr\"odinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.Comment: accepted for publication in M3A

Splitting schemes for the semi-linear wave equation with dynamic boundary conditions

This paper introduces novel splitting schemes of first and second order for the wave equation with kinetic and acoustic boundary conditions of semi-linear type. For kinetic boundary conditions, we propose a reinterpretation of the system equations as a coupled system. This means that the bulk and surface dynamics are modeled separately and connected through a coupling constraint. This allows the implementation of splitting schemes, which show first-order convergence in numerical experiments. On the other hand, acoustic boundary conditions naturally separate bulk and surface dynamics. Here, Lie and Strang splitting schemes reach first- and second-order convergence, respectively, as we reveal numerically

Regulation and technology innovation: A comparison of stated and formal regulatory barriers throughout the technology innovation process

Regulation is often mentioned as a barrier to technology innovation in various industries. Delayed market entry, stifled creativity, added activities and resource requirements are some frequently mentioned barriers. The study presented here explored various claims of regulation acting as a barrier to technology innovation. The findings suggest that formal statutory requirements only partly explain why regulation is perceived as a technology innovation barrier. Findings further indicate several discrepancies between stated and formal regulatory barriers and suggest that the majority of the stated barriers emerge within the organization during operationalization and the technology innovation process

On the port-Hamiltonian structure of the Navier-Stokes equations for reactive flows

We consider the problem of finding an energy-based formulation of the Navier-Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper we introduce a port-Hamiltonian formulation of the one-dimensional Navier-Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary.DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamic

P1-Nonconforming finite elements on triangulations into triangles and quadrilaterals

The P1-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park–Sheen and the Crouzeix–Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions