Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd.

Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. This Review Article outlines the exploration of the interplay between parity–time symmetry and non-Hermitian physics in optics, plasmonics and optomechanics.

Non-Hermitian physics and PT symmetry

This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed.


Keywords:

contact mechanics;
friction;
penalty method;
Lagrange multiplier method;
contact algorithms;
finite element method;
finite deformations;
discretization methods

Computational Contact Mechanics

In this paper, the algorithms of the automatic mesh generator NETGEN are described. The domain is provided by a Constructive Solid Geometry (CSG). The whole task of 3D mesh generation splits into four subproblems of special point calculation, edge following, surface meshing and finally volume mesh generation. Surface and volume mesh generation are based on the advancing front method. Emphasis is given to the abstract structure of the element generation rules. Several techniques of mesh optimization are tested and quality plots are presented.

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NETGEN An advancing front 2D/3D-mesh generator based on abstract rules

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.

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Reversing the pump dependence of a laser at an exceptional point

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.

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Equilibrated residual error estimator for edge elements

High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows

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.

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Joachim Schöberl

Papers

NETGEN An advancing front 2D/3D-mesh generator based on abstract rules

Reversing the pump dependence of a laser at an exceptional point

Equilibrated residual error estimator for edge elements

High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows

A posteriori error estimates for Maxwell equations