@misc{DelourmeSchmidtSemin, author = {Delourme, B{\´e}rang{\`e}re and Schmidt, Kersten and Semin, Adrien}, title = {On the homogenization of thin perforated walls of finite length}, series = {Asymptotic Analysis}, volume = {97}, journal = {Asymptotic Analysis}, number = {3-4}, issn = {0921-7134}, doi = {10.3233/ASY-151350}, pages = {211 -- 264}, abstract = {The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.}, language = {en} } @misc{SchmidtSemin, author = {Schmidt, Kersten and Semin, Adrien}, title = {Absorbing boundary conditions for the viscous acoustic wave equation}, series = {Mathemtatical Methods in the Applied Sciences}, volume = {39}, journal = {Mathemtatical Methods in the Applied Sciences}, number = {17}, issn = {1099-1476}, doi = {10.1002/mma.3755}, pages = {5043 -- 5065}, abstract = {We consider different acoustic models with viscosity in a semi-infinite waveguide with rigid walls, for which we propose and analyse absorbing boundary conditions on a truncated subdomain. The considered models are (i) the viscous acoustic equations in a stagnant mean flow, which exhibit for small viscosities boundary layers on the infinite walls, (ii) the limit equations for vanishing viscosity and (iii) a first-order approximation for low viscosity. The limit model (i) is well known as the Helmholtz equation for the pressure with homogeneous Neumann boundary conditions. For each of these models, the absorbing conditions appear as Dirichlet-to-Neumann (DtN) maps. The DtN boundary conditions for the singularly perturbed model (i) and the approximative model (iii) tend to the DtN boundary conditions of the limit problem (ii) if the viscosity approaches zero, and, hence, provide a uniform accuracy in the viscosity. The convergence of truncated DtN boundary conditions and the behaviour for viscosities tending to zero are shown in numerical experiments.}, language = {en} } @misc{FlissKlindworthSchmidt, author = {Fliss, Sonia and Klindworth, Dirk and Schmidt, Kersten}, title = {Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides}, series = {BIT Numerical Mathematics}, volume = {55}, journal = {BIT Numerical Mathematics}, number = {1}, issn = {0006-3835}, doi = {10.1007/s10543-014-0521-1}, pages = {81 -- 115}, abstract = {The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.}, language = {en} } @misc{ThoensZuevaSeminSchmidt, author = {Th{\"o}ns-Zueva, Anastasia and Semin, Adrien and Schmidt, Kersten}, title = {Multiharmonic analysis for nonlinear acoustics with different scales}, series = {ArXiv.org}, journal = {ArXiv.org}, pages = {20}, abstract = {The acoustic wave-propagation without mean flow and heat flux can be described in terms of velocity and pressure by the compressible nonlinear Navier-Stokes equations, where boundary layers appear at walls due to the viscosity and a frequency interaction appears, i.e. sound at higher harmonics of the excited frequency ω is generated due to nonlinear advection. We use the multiharmonic analysis to derive asymptotic expansions for small sound amplitudes and small viscosities both of order ε2 in which velocity and pressure fields are separated into far field and correcting near field close to walls and into contributions to the multiples of ω. Based on the asymptotic expansion we present approximate models for either the pressure or the velocity for order 0, 1 and 2, in which impedance boundary conditions include the effect of viscous boundary layers and contributions at frequencies 0 and 2⋅ω depend nonlinearly on the approximation at frequency ω. In difference to the Navier-Stokes equations in time domain, which has to be resolved numerically with meshes adaptively refined towards the wall boundaries and explicit schemes require the use of very small time steps, the approximative models can be solved in frequency domain on macroscopic meshes. We studied the accuracy of the approximated models of different orders in numerical experiments comparing with reference solutions in time-domain.}, language = {en} } @misc{SchmidtDiazHeier, author = {Schmidt, Kersten and Diaz, Julien and Heier, Christian}, title = {Non-conforming Galerkin finite element methods for local absorbing boundary conditions of higher order}, series = {Computers \& Mathematics with Applications}, volume = {70}, journal = {Computers \& Mathematics with Applications}, number = {9}, issn = {0898-1221}, doi = {10.1016/j.camwa.2015.08.034}, pages = {2252 -- 2269}, abstract = {A new non-conforming finite element discretization methodology for second order elliptic partial differential equations involving higher order local absorbing boundary conditions in 2D and 3D is proposed. The novelty of the approach lies in the application of C⁰-continuous finite element spaces, which is the standard discretization of second order operators, to the discretization of boundary differential operators of order four and higher. For each of these boundary operators, additional terms appear on the bound-ary nodes in 2D and on the boundary edges in 3D, similarly to interior penalty discontinuous Galerkin methods, which leads to a stable and consistent formulation. In this way, no auxiliary variables on the boundary have to be introduced and trial and test functions of higher smoothness along the boundary are not required. As a consequence, the method leads to lower computational costs for discretizations with higher order elements and is easily integrated in high-order finite element libraries. A priori h-conver-gence error estimates show that the method does not reduce the order of convergence compared to usual Dirichlet, Neumann or Robin boundary conditions if the polynomial degree on the boundary is increased simultaneously. A series of numerical experiments illustrates the utility of the method and validates the theoretical convergence results.}, language = {en} } @techreport{SchmidtSeminDelourme, author = {Schmidt, Kersten and Semin, Adrien and Delourme, B{\´e}rang{\`e}re}, title = {On the homogenization of the Helmholtz problem with thin perforated walls of finite length}, pages = {12}, abstract = {On the homogenization of the Helmholtz problem with thin perforated walls of finite length}, language = {en} } @techreport{SchmidtSeminDelourme, author = {Schmidt, Kersten and Semin, Adrien and Delourme, B{\´e}rang{\`e}re}, title = {When a thin periodic layer meets corners: asymptotic analysis of a singular Poisson problem}, pages = {58}, abstract = {The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization, and a complete justification is included in the paper or its appendix.}, language = {en} } @techreport{BoettgerFelgentreffHesseetal., author = {B{\"o}ttger, Stefan and Felgentreff, Bernd and Hesse, Gerold and Hloucal, Martin-Joseph and Leßmann, Dieter and Mix, Sebastian and Roselt, Kersten and Safarik, Mathias and Schmidt, J{\"o}rg and Steffan, Christoph and Uhlmann, Wilfried}, title = {Seethermie - Innovative W{\"a}rmeversorgung aus Tagebaurestseen - Schlussbericht}, address = {Leipzig}, pages = {146}, language = {de} }