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Auxiliary Space Preconditioners for SIP-DG Discretizations of H(curl)-elliptic Problems with Discontinuous Coefficients
(2015)Research ReportWe propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric Interior Penalty Discontinuous Galerkin (IP-DG) discretization of H(curl, )- elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners relies on the auxiliary space method (ASM) employing an auxiliary space of H(curl, )- conforming finite element functions together with ...Report -
Stabilized Galerkin for Transient Advection of Differential Forms
(2015)Research ReportWe deal with the discretization of generalized transient advection problems for differential forms on bounded spatial domains. We pursue an Eulerian method of lines approach with explicit time-stepping. Concerning spatial discretization we extend the jump stabilized Galerkin discretization proposed in [H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, Math. Modelling Numer. Analysis, 47 (2013), pp. 1713–1732] ...Report -
Large deformation shape uncertainty quantification in acoustic scattering
(2015)Research ReportWe address shape uncertainty quantification for the two-dimensional Helmholtz trans- mission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate ...Report -
Scatterers on the substrate: Far field formulas
(2015)Research ReportThe near-field to far-field mapping is a tool used to describe radiation at far distances from scatterers. We consider the geometric setting of a bounded scatterer mounted on a substrate, illuminated by a monochromatic plane wave. For such an structure, the far-field functional consists of different asymptotic terms including surface waves. We investigate all contributions closely and show that the only important term at far distances is ...Report -
A Survey of Trefftz Methods for the Helmholtz Equation
(2015)SAM Research ReportTrefftz 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 ...Report -
Integral Equations for Electromagnetic Scattering at Multi-Screens
(2015)Research ReportIn [X. Claeys and R. Hiptmair, Integral equations on multi-screens. Integral Equations and Operator Theory, 77(2):167–197, 2013] we developed a framework for the analysis of boundary integral equations for acoustic scattering at so-called multi-screens, which are arbitrary arrangements of thin panels made of impenetrable material. In this article we extend these considerations to boundary integral equations for electromagnetic scattering. ...Report -
Data Sparse Numerical Models for SNOM Tips
(2015)Research ReportWe propose matrix compression for the efficient numerical modeling of geometrically persistent parts in large-scale electromagnetic simulations. Our approach relies on local low-rank representation in the framework of the H-matrix data sparse matrix storage format.We discuss two ways to build approximate Hmatrix representations of dense Schur-complement matrices: Adaptive cross approximation (ACA) and Harithmetics. We perform profound ...Report -
Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects
(2015)Research ReportWe consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We start from the second-kind boundary integral approach of [X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33-57, 2015] ...Report