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Lutz Angermann

Publications and source records attributed to Lutz Angermann.

8 recordsLinked to original sources
arXiv2023

Finite element solution of a radiation/propagation problem for a Helmholtz equation with a compactly supported nonlinearity

A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model consists of a nonlinear Helmholtz equation that is reduced to a spherical domain. As a specific example, we consider a finite element method consisting of Courant-type elements with curved edges at the boundary of a circular computational domain in the two-dimensional case. We examine this method and more general conforming methods -- including three-dimensional ones -- with comparable properties for their well-posedness; in particular, the validity of a discrete inf-sup condition of the modified sesquilinear form uniformly with respect to both the truncation and the mesh parameters is shown. Under suitable assumptions to the nonlinearities, a quasi-optimal error estimate is obtained. Finally, the satisfiability of the approximation property of the finite element space required for the solvability of a class of adjoint linear problems is discussed.

math.NA↗
arXiv2023

An Energy Stable Discontinuous Galerkin Time-Domain Finite Element Method in Optics and Photonics

In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell's equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with $Q_k$-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy. The approaches presented prove to be robust and allow the modeling of optical problems and the treatment of complex nonlinearities as well as geometries of various physical systems coupled with electromagnetic fields.

math.NA↗
arXiv2023

A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

The present work describes some extensions of an approach, originally developed by V.V. Yatsyk and the author, for the theoretical and numerical analysis of scattering and radiation effects on infinite plates with cubically polarized layers. The new aspects lie on the transition to more generally shaped, two- or three-dimensional objects, which no longer necessarily have to be represented in terms a Cartesian product of real intervals, to more general nonlinearities (including saturation) and the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.). The paper advocates an approach that consists in transforming the original full-space problem for a nonlinear Helmholtz equation (as the simplest model) into an equivalent boundary-value problem on a bounded domain by means of a nonlocal Dirichlet-to-Neumann (DtN) operator. It is shown that the transformed problem is equivalent to the original one and can be solved uniquely under suitable conditions. Morever, the impact of the truncation of the DtN operator on the resulting solution is investigated, so that the way to the numerical solution by appropriate finite element methods is available.

math-ph↗
arXiv2023

Error estimates for completely discrete FEM in energy-type and weaker norms

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion-convection-reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy-type norms and, by generalizing the Aubin-Nitsche technique, optimal error estimates in weaker norms.

math.NA↗
arXiv2012

A posteriori estimates for errors of functionals on finite volume approximations to solutions of elliptic boundary value problems

This article describes the extension of recent methods for a posteriori error estimation such as dual-weighted residual methods to node-centered finite volume discretizations of second order elliptic boundary value problems including upwind discretizations. It is shown how different sources of errors, in particular modeling errors and discretization errors, can be estimated with respect to a user-defined output functional.

math.NA↗
arXiv2011

Interpolation, projection and hierarchical bases in discontinuous Galerkin methods

The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities. The main focus is directed to applications to discrete conservation laws.

math.NA↗