TY - GEN A1 - Schädle, Achim A1 - Zschiedrich, Lin T1 - Additive Schwarz method for scattering problems using the PML method at interfaces N2 - Scattering problems in integrated optics can be modeled in simple cases by the Helmholtz equation. The computational domain is truncated by a non-reflecting boundary condition. We investigate Schwarz algorithms with a sort of DtN operator, realized by the PML-method, at the interfaces of the sub-domains as an iterative solver. T3 - ZIB-Report - 05-27 KW - perfectly matched layer KW - Helmholtz equation KW - scattering KW - domain decomposition Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8613 ER - TY - GEN A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian A1 - Schädle, Achim T1 - Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory N2 - To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing $N$ steps requires only $O(N\log N)$ operations and $O(\log N)$ active memory, in place of $O(N^2)$ operations and $O(N)$ memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law. T3 - ZIB-Report - 06-45 KW - convolution quadrature KW - adaptivity KW - Volterra integral equations KW - numerical inverse Laplace transform KW - anomalous diffusion KW - fractional order visco Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9399 ER - TY - GEN A1 - Schädle, Achim A1 - Zschiedrich, Lin A1 - Burger, Sven A1 - Klose, Roland A1 - Schmidt, Frank T1 - Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures N2 - We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed. We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer. T3 - ZIB-Report - 06-04 KW - domain decomposition KW - conical diffraction KW - electro-magnetic scattering KW - Maxwell's equations KW - Lithography KW - finite elements Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8984 ER - TY - GEN A1 - Lopez-Fernandez, Maria A1 - Palencia, Cesar A1 - Schädle, Achim T1 - A spectral order method for inverting sectorial Laplace transforms N2 - Laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered. A spectral order, parallelizable method for their numerical inversion is proposed. The method takes into account the available information about the errors arising in the evaluations. Several numerical illustrations are provided. T3 - ZIB-Report - 05-26 KW - Laplace transform KW - numerical inversion KW - parabolic KW - spectral order KW - parallelizable Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8607 ER - TY - JOUR A1 - Hiptmair, Ralf A1 - Schädle, Achim T1 - Non-reflecting boundary conditions for Maxwell’s equations JF - Computing Y1 - 2003 UR - http://dx.doi.org/10.1007/s00607-003-0026-2 VL - 71 IS - 3 ER - TY - JOUR A1 - Dolling, G. A1 - Klein, M. A1 - Wegener, M. A1 - Schädle, Achim A1 - Burger, Sven A1 - Linden, S. T1 - Negative beam displacement for negative-index photonic metamaterials JF - Optics Express Y1 - 2007 U6 - https://doi.org/10.1364/OE.15.014219 VL - 15 IS - 21 SP - 14219 EP - 14227 ER - TY - JOUR A1 - Dolling, G. A1 - Wegener, M. A1 - Schädle, Achim A1 - Burger, Sven A1 - Linden, S. T1 - Observation of magnetization waves in negative-index photonic metamaterials JF - Appl. Phys. Lett. Y1 - 2006 U6 - https://doi.org/10.1063/1.2403905 VL - 89 SP - 231118 ER - TY - JOUR A1 - López-Fernández, Maria A1 - Lubich, Christian A1 - Palencia, Cesar A1 - Schädle, Achim T1 - Fast Runge-Kutta approximation of inhomogeneous parabolic equations JF - Numer. Math. Y1 - 2005 U6 - https://doi.org/http://dx.doi.org/10.1007/s00211-005-0624-3 VL - 102 SP - 277 EP - 291 ER - TY - JOUR A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian A1 - Schädle, Achim T1 - Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory JF - SIAM J. Sci. Comput. Y1 - 2008 U6 - https://doi.org//10.1137/060674168 VL - 30 IS - 2 SP - 1015 EP - 1037 ER - TY - JOUR A1 - Lopez-Fernandez, Maria A1 - Palencia, Cesar A1 - Schädle, Achim T1 - A spectral order method for inverting sectorial Laplace transforms JF - SIAM J. Numer. Anal. Y1 - 2006 U6 - https://doi.org//10.1137/050629653 VL - 44 IS - 3 SP - 1332 EP - 1350 ER - TY - JOUR A1 - Kalkbrenner, T. A1 - Hakanson, U. A1 - Schädle, Achim A1 - Burger, Sven A1 - Henkel, C. A1 - Sandoghdar, V. T1 - Optical microscopy using the spectral modifications of a nano-antenna JF - Phys. Rev. Lett. Y1 - 2005 U6 - https://doi.org/10.1103/PhysRevLett.95.200801 VL - 95 SP - 200801 ER - TY - CHAP A1 - Schädle, Achim A1 - Zschiedrich, Lin ED - Widlund, Olof ED - Keyes, David T1 - Additive Schwarz method for scattering problems using the PML method at interfaces T2 - Domain Decomposition Methods in Science and Engineering XVI Y1 - 2007 VL - 55 SP - 205 EP - 212 PB - Springer ER - TY - JOUR A1 - Schädle, Achim A1 - Zschiedrich, Lin A1 - Burger, Sven A1 - Klose, Roland A1 - Schmidt, Frank T1 - Domain Decomposition Method for Maxwell’s Equations JF - J. Comput. Phys. Y1 - 2007 U6 - https://doi.org/10.1016/j.jcp.2007.04.017 VL - 226 SP - 477 EP - 493 ER - TY - JOUR A1 - Schmidt, Frank A1 - Hohage, Thorsten A1 - Klose, Roland A1 - Schädle, Achim A1 - Zschiedrich, Lin T1 - Pole condition: A numerical method for Helmholtz-type scattering problems with inhomogeneous exterior domain JF - J. Comput. Appl. Math. Y1 - 2008 U6 - https://doi.org/10.1016/j.cam.2007.04.046 VL - 218 IS - 1 SP - 61 EP - 69 ER - TY - JOUR A1 - Ruprecht, Daniel A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Transparent boundary conditions for time-dependent problems JF - SIAM J. Sci. Comput. Y1 - 2008 U6 - https://doi.org/10.1137/070692637 VL - 30 IS - 5 SP - 2358 EP - 2385 ER - TY - JOUR A1 - Schädle, Achim A1 - López-Fernández, María A1 - Lubich, Christian T1 - Fast and oblivious convolution quadrature JF - SIAM J. Sci. Comput. Y1 - 2006 U6 - https://doi.org/http://dx.doi.org/10.1137/050623139 VL - 28 IS - 2 SP - 421 EP - 438 ER - TY - CHAP A1 - Schädle, Achim A1 - Ruprecht, Daniel A1 - Schmidt, Frank ED - et.al., N. Biggs T1 - Transparent boundary conditions - the pole condition approach T2 - Proceedings of Waves 2007 Y1 - 2007 SP - 301 EP - 303 ER - TY - CHAP A1 - Schmidt, Frank A1 - Burger, Sven A1 - Schädle, Achim A1 - Klose, Roland T1 - Finite element methods for optical device design T2 - Proceedings of the 5th International Conference on Numerical Simulation of Optoelectronic devices Y1 - 2005 U6 - https://doi.org/10.1109/NUSOD.2005.1518133 SP - 57 EP - 58 ER - TY - CHAP A1 - Zschiedrich, Lin A1 - Burger, Sven A1 - Klose, Roland A1 - Schädle, Achim A1 - Schmidt, Frank ED - Sidorin, Y. ED - Wächter, Christoph T1 - JCMmode: an adaptive finite element solver for the computation of leaky modes T2 - Integrated Optics: Devices, Materials, and Technologies IX Y1 - 2005 U6 - https://doi.org/10.1117/12.590372 VL - 5728 SP - 192 EP - 202 PB - Proc. SPIE ER - TY - CHAP A1 - Zschiedrich, Lin A1 - Burger, Sven A1 - Schädle, Achim A1 - Schmidt, Frank ED - Levinson, Harry ED - Dusa, Mircea T1 - A rigorous finite-element domain decomposition method for electromagnetic near field simulations T2 - Optical Microlithography XXI Y1 - 2008 U6 - https://doi.org/10.1117/12.771989 VL - 6924 PB - SPIE ER - TY - CHAP A1 - Zschiedrich, Lin A1 - Burger, Sven A1 - Schädle, Achim A1 - Schmidt, Frank T1 - Domain decomposition method for electromagnetic scattering problems: application to EUV lithography T2 - Proceedings of the 5th International Conference on Numerical Simulation of Optoelectronic devices Y1 - 2005 U6 - https://doi.org/10.1109/NUSOD.2005.1518132 SP - 55 EP - 56 ER - TY - JOUR A1 - Zschiedrich, Lin A1 - Klose, Roland A1 - Schädle, Achim A1 - Schmidt, Frank T1 - A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D JF - J. Comput. Appl. Math. Y1 - 2006 U6 - https://doi.org/10.1016/j.cam.2005.03.047 VL - 188 SP - 12 EP - 32 ER - TY - GEN A1 - Burger, Sven A1 - Klose, Roland A1 - Schädle, Achim A1 - Zschiedrich, Lin T1 - HelmPole - A finite element solver for scattering problems on unbounded domains: Implementation based on PML N2 - The solution of scattering problems described by the Helmholtz equation on unbounded domains is of importance for a wide variety of applications, for example in electromagnetics and acoustics. An implementation of a solver for scattering problems based on the programming language Matlab is introduced. The solver relies on the finite-element-method and on the perfectly-matched-layer-method, which allows for the simulation of scattering problems on complex geometries surrounded by inhomogeneous exterior domains. This report gives a number of detailed examples and can be understood as a user manual to the freely accessible code of the solver HelmPole. T3 - ZIB-Report - 03-38 KW - Helmholtz equation KW - finite element method KW - perfectly matched layer Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7609 ER - TY - GEN A1 - Zschiedrich, Lin A1 - Klose, Roland A1 - Schädle, Achim A1 - Schmidt, Frank T1 - A new Finite Element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D N2 - In this paper we propose a new finite element realization of the Perfectly Matched Layer method (PML-method). Our approach allows to deal with arbitrary shaped polygonal domains and with certain types of inhomogeneous exterior domains. Among the covered inhomogeneities are open waveguide structures playing an essential role in integrated optics. We give a detailed insight to implementation aspects. Numerical examples show exponential convergence behavior to the exact solution with the thickness of the PML sponge layer. T3 - ZIB-Report - 03-44 KW - transparent boundary conditions KW - perfectly matched layer KW - pole condition Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7662 ER - TY - GEN A1 - Schädle, Achim A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian T1 - Fast and oblivious convolution quadrature N2 - We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N\, \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integro-differential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions. T3 - ZIB-Report - 05-09 KW - convolution KW - numerical integration KW - Runge-Kutta methods KW - Volterra integral equation KW - anomalous diffusion Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8437 ER - TY - GEN A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian A1 - Palencia, Cesar A1 - Schädle, Achim T1 - Fast Runge-Kutta approximation of inhomogeneous parabolic equations N2 - The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\varepsilon$, by solving only $$O\Big(\log N\, \log \frac1\varepsilon \Big) $$ linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm. T3 - ZIB-Report - 05-10 KW - parabolic equation KW - Runge-Kutta methods Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8443 ER - TY - JOUR A1 - Antoine, Xavier A1 - Arnold, Anton A1 - Besse, Christophe A1 - Ehrhardt, Matthias A1 - Schädle, Achim T1 - A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations JF - Commun. Comput. Phys. Y1 - 2008 UR - http://www.global-sci.com/openaccess/v4_729.pdf VL - 4 SP - 729 EP - 796 ER - TY - CHAP A1 - Burger, Sven A1 - Zschiedrich, Lin A1 - Klose, Roland A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Enkrich, Christian A1 - Linden, S. A1 - Wegener, M. A1 - Soukoulis, C. ED - Szoplik, T. ED - Özbay, E. ED - Soukoulis, C. ED - Zheludev, N. T1 - Numerical Investigation of Light Scattering off Split-Ring Resonators T2 - Metamaterials Y1 - 2005 U6 - https://doi.org/10.1117/12.622184 VL - 5955 SP - 18 EP - 26 PB - Proc. SPIE ER - TY - CHAP A1 - Burger, Sven A1 - Zschiedrich, Lin A1 - Schädle, Achim A1 - Klose, Roland A1 - Kettner, Benjamin A1 - Pomplun, Jan A1 - Pollok, T. A1 - Schmidt, Frank T1 - Advanced FEM Analysis of Nano-Optical Devices T2 - Proc. Symposium on Photonics Technologies for the 7th Framework Programme Y1 - 2006 ER - TY - CHAP A1 - Burger, Sven A1 - Schädle, Achim A1 - Zschiedrich, Lin A1 - Schmidt, Frank ED - Andrews, David L. ED - Nunzi, Jean-Michel ED - Ostendorf, Andreas T1 - Advanced FEM Analysis of Nano-Optical Devices T2 - Nanophotonics Y1 - 2006 U6 - https://doi.org/10.1117/12.662942 VL - 6195 SP - 72 EP - 82 PB - Proc. SPIE ER - TY - CHAP A1 - Burger, Sven A1 - Klose, Roland A1 - März, Reinhard A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Efficient Finite Element Methods for the Design of Microoptical Components T2 - Proc. Microoptics Conf. 2004 Y1 - 2004 ER - TY - CHAP A1 - Burger, Sven A1 - Klose, Roland A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Zschiedrich, Lin ED - Sidorin, Y. ED - Wächter, Christoph T1 - FEM modelling of 3D photonic crystals and photonic crystal waveguides T2 - Integrated Optics : Devices, Materials and Technologies IX Y1 - 2005 U6 - https://doi.org/10.1117/12.585895 VL - 5728 SP - 164 EP - 173 PB - Proc. SPIE ER - TY - CHAP A1 - Burger, Sven A1 - Klose, Roland A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Adaptive FEM solver for the computation of electromagnetic eigenmodes in 3D photonic crystal structures T2 - Scientific Computing in Electrical Engineering Y1 - 2006 U6 - https://doi.org/10.1007/978-3-540-32862-9_24 SP - 169 EP - 175 PB - Springer Verlag ER - TY - JOUR A1 - Ruprecht, Daniel A1 - Schädle, Achim A1 - Schmidt, Frank T1 - Transparent boundary conditions based on the pole condition for time-dependent, two-dimensional problems JF - Numerical Methods for Partial Differential Equations Y1 - 2013 U6 - https://doi.org/10.1002/num.21759 VL - 29 IS - 4 SP - 1367 EP - 1390 ER - TY - GEN A1 - Antoine, Xavier A1 - Arnold, Anton A1 - Besse, Christophe A1 - Ehrhardt, Matthias A1 - Schädle, Achim T1 - A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations N2 - In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case. T3 - ZIB-Report - 07-34 KW - Schrödinger equation KW - transparent boundary condition KW - discrete convolution KW - unbounded domain Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10332 SN - 1438-0064 ER - TY - GEN A1 - Ruprecht, Daniel A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Transparent boundary conditons for time-dependent problems N2 - A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients. T3 - ZIB-Report - 07-12 KW - transparent boundary condition KW - non-reflecting boundary condition KW - pole condition KW - wave equation KW - Schrödinger equation KW - drift diffusion equation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9558 ER -