Fast and stable contour integration for high order divided differences via elliptic functions

In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator valued functions. This is a classical problem in numerical mathematics but also arises in new applications such as, e.g., the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half plane, polynomially to the right half plane and have an oscillatory behaviour with increasing imaginary part. The interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

A Refined Finite Element Convergence Theory for Highly Indefinite Helmholtz Problems

Abstract.: It is well known that standard h-version finite element discretisations using lowest order elements for Helmholtz' equation suffer from the following stability condition: 's'sThe mesh width h of the finite element mesh has to satisfy k 2 h≲1'', where k denotes the wave number. This condition rules out the reliable numerical solution of Helmholtz equation in three dimensions for large wave numbers k≳50. In our paper, we will present a refined finite element theory for highly indefinite Helmholtz problems where the stability of the discretisation can be checked through an 's'salmost invariance'' condition. As an application, we will consider a one-dimensional finite element space for the Helmholtz equation and apply our theory to prove stability under the weakened condition hk≲1 and optimal convergence estimate

"It's the real thing": performance and murder in Sweden.

The article investigates contemporary experimental theatre in Sweden. It sums up and probes the implications of Sju tre (1999), the most controversial theatre production in Sweden in modern times. Lars Nor'n, the playwright and director, staged a dialogue involving three real convicts, of whom two were outspoken Nazis. The article explores the uncertain boundaries between aesthetic, ethical, and political issues with ramifications regarding the wider public opinion in Sweden, on racism and crime. It is methodologically motivated by reception research, performativity and idealogical discourse. By virtue of its performative impact, the theatrical event proved to be directly linked with critical questions of democracy, although conceivably at the expense of the artistic integrity of the director and the theatre as creator of public opinion. The article points to a paradox of democracy whereby hate speech is at once allowed and unjustified in the theatre as national arena. The actors are described and analysed as parasites in a societal body, that in Sju tre, becomes politically epitomised

Probing the gluon density of the proton in the exclusive photoproduction of vector mesons at the LHC: A phenomenological analysis

The current uncertainty on the gluon density extracted from the global parton analysis is large in the kinematical range of small values of the Bjorken - $x$ variable and low values of the hard scale $Q^2$. An alternative to reduces this uncertainty is the analysis of the exclusive vector meson photoproduction in photon - hadron and hadron - hadron collisions. This process offers a unique opportunity to constrain the gluon density of the proton, since its cross section is proportional to the gluon density squared. In this paper we consider current parametrizations for the gluon distribution and estimate the exclusive vector meson photoproduction cross section at HERA and LHC using the leading logarithmic formalism. We perform a fit of the normalization of the $\gamma h$ cross section and the value of the hard scale for the process and demonstrate that the current LHCb experimental data are better described by models that assume a slow increasing of the gluon distribution at small - $x$ and low $Q^2$.Comment: 8 pages, 6 figures, 1 table. Version published in European Physical Journal

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

Many important physical applications are governed by the wave equation. The formulation as time domain boundary integral equations involves retarded potentials. For the numerical solution of this problem, we employ the convolution quadrature method for the discretization in time and the Galerkin boundary element method for the space discretization. We introduce a simple a priori cut-off strategy where small entries of the system matrices are replaced by zero. The threshold for the cut-off is determined by an a priori analysis which will be developed in this paper. This analysis will also allow to estimate the effect of additional perturbations such as panel clustering and numerical integration on the overall discretization error. This method reduces the storage complexity for time domain integral equations from O(M2N) to O(M2N½ logM), where N denotes the number of time steps and M is the dimension of the boundary element spac

Damping of electromagnetic waves due to electron-positron pair production

The problem of the backreaction during the process of electron-positron pair production by a circularly polarized electromagnetic wave propagating in a plasma is investigated. A model based on the relativistic Boltzmann-Vlasov equation with a source term corresponding to the Schwinger formula for the pair creation rate is used. The damping of the wave, the nonlinear up-shift of its frequency due to the plasma density increase and the effect of the damping on the wave polarization and on the background plasma acceleration are investigated as a function of the wave amplitude.Comment: 11 pages, 5 figures; revtex

Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE-MG

Retarded boundary integral equations on the sphere: exact and numerical solution

In this paper we consider the three-dimensional wave equation in unbounded domains with Dirichlet boundary conditions. We start from a retarded single-layer potential ansatz for the solution of these equations which leads to the retarded potential integral equation on the bounded surface of the scatterer. We formulate an algorithm for the space-time Galerkin discretization with smooth and compactly supported temporal basis functions, which were introduced in Sauter & Veit (2013, Numer. Math., 145-176). For the debugging of an implementation and for systematic parameter tests it is essential to have at hand some explicit representations and some analytic properties of the exact solutions for some special cases. We will derive such explicit representations for the case where the scatterer is the unit ball. The obtained formulas are easy to implement and we will present some numerical experiments for these cases to illustrate the convergence behaviour of the proposed metho