Joachim Schöberl

Reversing the pump dependence of a laser at an exceptional point.

When two resonant modes in a system with gain or loss coalesce in both their resonance position and their width, a so-called exceptional point occurs, which acts as a source of non-trivial physics in a diverse range of systems. Lasers provide a natural setting to study such non-Hermitian degeneracies, as they feature resonant modes and a gain material as their basic constituents. Here we show that exceptional points can be conveniently induced in a photonic molecule laser by a suitable variation of the applied pump. Using a pair of coupled microdisk quantum cascade lasers, we demonstrate that in the vicinity of these exceptional points the coupled laser shows a characteristic reversal of its pump dependence, including a strongly decreasing intensity of the emitted laser light for increasing pump power.

Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block systems is the positivity of the (1,1) block on the kernel of the (2,1) block. Many solution methods, however, require that the positivity of the (1,1) block is satisfied everywhere. To enforce the positivity everywhere, an augmented Lagrangian approach is usually chosen. However, the adjustment of the involved parameters is a critical issue. We will present a different approach that is not based on such an explicit augmentation technique. For the considered class of symmetric and indefinite preconditioners, assumptions are presented that lead to symmetric and positive definite problems with respect to a particular scalar product. Therefore, conjugate gradient acceleration can be used. An important class of applications are optimal control problems. It is typical for such problems that the cost functional contains an extra regularization parameter. For control problems with elliptic state equations and distributed control, a special indefinite preconditioner for the discretized problem is constructed, which leads to convergence rates of the preconditioned conjugate gradient method that are not only independent of the mesh size but also independent of the regularization parameter. Numerical experiments are presented for illustrating the theoretical results.

A posteriori error estimates for Maxwell equations

Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nedelec finite elements. In Beck et al., 2000, a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasi-interpolation operators recently introduced in J. Schoberl, Commuting quasi-interpolation operators for mixed finite elements. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.

An algebraic multigrid method for finite element discretizations with edge elements

This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,Ω). The finite element spaces are generated by Nedelecs edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright

Equilibrated residual error estimator for edge elements

Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to the curl-curl equation and edge elements. The construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. For this reason, Raviart-Thomas elements are extended in the spirit of distributions.

High order Nedelec elements with local complete sequence properties

Purpose – The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements.Design/methodology/approach – Discusses a systematic strategy for the realization of arbitrary order hierarchic H(curl)‐conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowest order Nedelec, higher‐order edge‐based, face‐based (only in 3D) and element‐based ones.Findings – Our new shape functions provide not only the global complete sequence property but also local complete sequence properties for each edge‐, face‐, and element‐block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block‐diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradien...

Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination

A new active set based algorithm is proposed that uses the conjugate gradient method to explore the face of the feasible region defined by the current iterate and the reduced gradient projection with the fixed steplength to expand the active set. The precision of approximate solutions of the auxiliary unconstrained problems is controlled by the norm of violation of the Karush-Kuhn-Tucker conditions at active constraints and the scalar product of the reduced gradient with the reduced gradient projection. The modifications were exploited to find the rate of convergence in terms of the spectral condition number of the Hessian matrix, to prove its finite termination property even for problems whose solution does not satisfy the strict complementarity condition, and to avoid any backtracking at the cost of evaluation of an upper bound for the spectral radius of the Hessian matrix. The performance of the algorithm is illustrated on solution of the inner obstacle problems. The result is an important ingredient in development of scalable algorithms for numerical solution of elliptic variational inequalities.

Finite-element simulation of wave propagation in periodic piezoelectric SAW structures

Many surface acoustic wave (SAW) devices consist of quasiperiodic structures that are designed by successive repetition of a base cell. The precise numerical simulation of such devices, including all physical effects, is currently beyond the capacity of high-end computation. Therefore, we have to restrict the numerical analysis to the periodic substructure. By using the finite-element method (FEM), this can be done by introducing periodic boundary conditions (PBCs) at special artificial boundaries. To be able to describe the complete dispersion behavior of waves, including damping effects, the PBC has to be able to model each mode that can be excited within the periodic structure. Therefore, the condition used for the PBCs must hold for each phase and amplitude difference existing at periodic boundaries. Based on the Floquet theorem, our two newly developed PBC algorithms allow the calculation of both, the phase and the amplitude coefficients of the wave. In the first part of this paper we describe the basic theory of the PBCs. Based on the FEM, we develop two different methods that deliver the same results but have totally different numerical properties and, therefore, allow the use of problem-adapted solvers. Further on, we show how to compute the charge distribution of periodic SAW structures with the aid of the new PBCs. In the second part, we compare the measured and simulated dispersion behavior of waves propagating on periodic SAW structures for two different piezoelectric substrates. Then we compare measured and simulated input admittances of structures similar to SAW resonators.

Crouzeix-Raviart type finite elements on anisotropic meshes

Summary. The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.

Solving the Signorini problem on the basis of domain decomposition techniques

The finite element discretization of the Signorini Problem leads to a large scale constrained minimization problem. To improve the convergence rate of the projection method preconditioning must be developed. To be effective, the relative condition number of the system matrix with respect to the preconditioning matrix has to be small and the applications of the preconditioner as well as the projection onto the set of feasible elements have to be fast computable. In this paper, we show how to construct and analyze such preconditioners on the basis of domain decomposition techniques. The numerical results obtained for the Signorini problem as well as for contact problems in plane elasticity confirm the theoretical analysis quite well.