On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Analytical investigations and numerical experiments for singularly perturbed convection-diffusion problems with layers and singularities using a newly developed FE-software

In the field of singularly perturbed reaction- or convection-diffusion boundary value problems the research area of a priori error analysis for the finite element method, has already been thoroughly investigated. In particular, for mesh adapted methods and/or various stabilization techniques, works have been done that prove optimal rates of convergence or supercloseness uniformly in the perturbation parameter epsilon. Commonly, however, it is assumed that the exact solution behaves nicely in that it obeys certain regularity assumptions although in general, e.g. due to corner singularities, these regularity requirements are not satisfied. So far, insufficient regularity has been met by assuming compatibility conditions on the data. The present thesis originated from the question: What can be shown if these rather unrealistic additional assumptions are dropped? We are interested in epsilon-uniform a priori estimates for convergence and superconvergence that include some regularity parameter that is adjustable to the smoothness of the exact solution. A major difficulty that occurs when seeking the numerical error decay is that the exact solution is not known. Since we strive for reliable rates of convergence we want to avoid the standard approach of the "double-mesh principle". Our choice is to use reference solutions as a substitute for the exact solution. Numerical experiments are intended to confirm the theoretical results and to bring further insights into the interplay between layers and singularities. To computationally realize the thereby arising demanding practical aspects of the finite element method, a new software is developed that turns out to be particularly suited for the needs of the numerical analyst. Its design, features and implementation is described in detail in the second part of the thesis

Applications of Asymptotic Analysis

This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and on other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods

Drift-diffusion models for innovative semiconductor devices and their numerical solution

We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization

Multiscale methods for the solution of the Helmholtz and Laplace equations

This paper presents some numerical results about applications of multiscale techniques to boundary integral equations. The numerical schemes developed here are to some extent based on the results of the papers [6]—[10]. Section 2 deals with a short description of the theory of generalized Petrov-Galerkin methods for elliptic periodic pseudodifferential equations in $\mathbb{R}^n$ covering classical Galerkin schemes, collocation, and other methods. A general setting of multiresolution analysis generated by periodized scaling functions as well as a general stability and convergence theory for such a framework is outlined. The key to the stability analysis is a local principle due to one of the authors. Its applicability relies here on a sufficiently general version of a so-called discrete commutator property of wavelet bases (see [6]). These results establish important prerequisites for developing and analysing methods for the fast solution of the resulting linear systems (Section 2.4). The crucial fact which is exploited by these methods is that the stiffness matrices relative to an appropriate wavelet basis can be approximated well by a sparse matrix while the solution to the perturbed problem still exhibits the same asymptotic accuracy as the solution to the full discrete problem. It can be shown (see [7]) that the amount of the overall computational work which is needed to realize a required accuracy is of the order $\mathcal{O}(N(\log N)^b)$, where $N$ is the number of unknowns and $b \geq 0$ is some real number

Computational Electromagnetism and Acoustics

The challenge inherent in the accurate and efficient numerical modeling of wave propagation phenomena is the common grand theme in both computational electromagnetics and acoustics. Many excellent contributions at this Oberwolfach workshop were devoted to this theme and a wide range of numerical techniques and algorithms were mustered to tackle this challenge

A survey of Trefftz methods for the Helmholtz equation

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes in Computational Science and Engineering. Differences from v1: added a few sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction