Transmission problems and domain decompositions for non-autonomous parabolic equations on evolving domains

Parabolic equations on evolving domains model a multitude of applications including various industrial processes such as the molding of heated materials. Such equations are numerically challenging as they require large-scale computations and the usage of parallel hardware. Domain decomposition is a common choice of numerical method for stationary domains, as it gives rise to parallel discretizations. In this study, we introduce a variational framework that extends the use of such methods to evolving domains. In particular, we prove that transmission problems on evolving domains are well posed and equivalent to the corresponding parabolic problems. This in turn implies that the standard non-overlapping domain decompositions, including the Robin-Robin method, become well defined approximations. Furthermore, we prove the convergence of the Robin?Robin method. The framework is based on a generalization of fractional Sobolev-Bochner spaces on evolving domains, time-dependent Steklov-Poincaré operators, and elements of the approximation theory for monotone maps

Stationary Apollonian packings

The notion of stationary Apollonian packings in the d-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for the growth durations and it is shown that the packing is space-filling with probability 1, in the sense that the Lebesgue measure of its complement is zero. Finally, the phenomenon is studied that grains arrange in clusters and properties related to percolation are investigated

Dimension reduction for a coupled electro-elastic saddle-point problem at finite strains

oai:archive.wias-berlin.de:wias_mods_00009610We study the finite deformation of a thin, elastically heterogeneous sheet subject to electrostatic coupling. The interaction between mechanics and electrostatics is formulated as a saddle-point problem involving the deformation and the electrostatic potential. Starting from a three-dimensional electro-elastic model with prestrain in the elastic energy, we rigorously derive a reduced plate model in the bending regime. To perform the dimension reduction, that is, to derive the energy of a thin object by taking a suitable limit as its thickness tends to zero, we apply Gamma-convergence-type methods to the underlying saddle-point problem. In the case of bivariate functionals, this convergence is understood in an adapted epi/hypo-convergence sense. In this concept, we demonstrate the convergence of the rescaled electro-elastic problems to an effective two-dimensional bending model coupled to electric effects. We verify that cluster points of saddle points are saddle points for the limit

Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability, and optimal control

Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau–Yosida-type penalisation for the QVI, wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result

Uniqueness and regularity of weak solutions of a drift-diffusion system for perovskite solar cells

We establish a novel uniqueness result for an instationary drift-diffusion model for perovskite solar cells. This model for vacancy-assisted charge transport uses Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite. Existence of weak solutions and their boundedness was proven in a previous work. For the uniqueness proof, we establish improved integrability of the gradients of the charge-carrier densities. Based on estimates obtained in the previous paper, we consider suitably regularized continuity equations with partly frozen arguments and apply the regularity results for scalar quasilinear elliptic equations by Meinlschmidt & Rehberg, Evolution Equations and Control Theory, 2016, 5(1):147-184

Multi-scale hybrid band simulation of (Al,Ga)N UV-C light emitting diodes

Aluminium gallium nitride alloys are used for developing light emitting diodes operating in the UV part of the electromagnetic spectrum. These devices suffer from a low efficiency. To gain insight to this question we develop a 3-D modified drift-diffusion model which takes into account both alloy disorder effects and valence band mixing, and investigate the device efficiency. Results show that the current injection efficiency is strongly influenced by the chosen doping profile

Hierarchical proximal Galerkin: A fast hp-FEM solver for variational problems with pointwise inequality constraints

We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024), a recently introduced mesh-independent algorithm, to obtain a high-order finite element solver for variational problems with pointwise inequality constraints. This is achieved by discretizing the saddle point systems, arising from the latent variable proximal point method, with the hierarchical p-finite element basis. This results in discretized sparse Newton systems that admit a simple and effective block preconditioner. The solver can handle both obstacle-type and gradient-type constraints. We apply the resulting algorithm to solve obstacle problems with hp-adaptivity, a gradient-type constrained problem, and the thermoforming problem, an example of an obstacle-type quasi-variational inequality. We observe hp-robustness in the number of Newton iterations and only mild growth in the number of inner Krylov iterations to solve the Newton systems. Crucially we also provide wall-clock timings that are faster than low-order discretization counterparts

Reference map approach to Eulerian thermomechanics using GENERIC

An Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordi- nates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e. the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated