@misc{SchaedleZschiedrich2005, author = {Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {Additive Schwarz method for scattering problems using the PML method at interfaces}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8613}, number = {05-27}, year = {2005}, abstract = {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.}, language = {en} } @misc{LopezFernandezLubichSchaedle2006, author = {Lopez-Fernandez, Maria and Lubich, Christian and Sch{\"a}dle, Achim}, title = {Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory}, doi = {10.1137/060674168}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9399}, number = {06-45}, year = {2006}, abstract = {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 Schr{\"o}dinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.}, language = {en} } @misc{SchaedleZschiedrichBurgeretal.2006, author = {Sch{\"a}dle, Achim and Zschiedrich, Lin and Burger, Sven and Klose, Roland and Schmidt, Frank}, title = {Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures}, doi = {/10.1016/j.jcp.2007.04.017}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8984}, number = {06-04}, year = {2006}, abstract = {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.}, language = {en} } @misc{LopezFernandezPalenciaSchaedle2005, author = {Lopez-Fernandez, Maria and Palencia, Cesar and Sch{\"a}dle, Achim}, title = {A spectral order method for inverting sectorial Laplace transforms}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8607}, number = {05-26}, year = {2005}, abstract = {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.}, language = {en} } @article{HiptmairSchaedle2003, author = {Hiptmair, Ralf and Sch{\"a}dle, Achim}, title = {Non-reflecting boundary conditions for Maxwell's equations}, volume = {71}, journal = {Computing}, number = {3}, year = {2003}, language = {en} } @article{DollingKleinWegeneretal.2007, author = {Dolling, G. and Klein, M. and Wegener, M. and Sch{\"a}dle, Achim and Burger, Sven and Linden, S.}, title = {Negative beam displacement for negative-index photonic metamaterials}, volume = {15}, journal = {Optics Express}, number = {21}, doi = {10.1364/OE.15.014219}, pages = {14219 -- 14227}, year = {2007}, language = {en} } @article{DollingWegenerSchaedleetal.2006, author = {Dolling, G. and Wegener, M. and Sch{\"a}dle, Achim and Burger, Sven and Linden, S.}, title = {Observation of magnetization waves in negative-index photonic metamaterials}, volume = {89}, journal = {Appl. Phys. Lett.}, doi = {10.1063/1.2403905}, pages = {231118}, year = {2006}, language = {en} } @article{LopezFernandezLubichPalenciaetal.2005, author = {L{\´o}pez-Fern{\´a}ndez, Maria and Lubich, Christian and Palencia, Cesar and Sch{\"a}dle, Achim}, title = {Fast Runge-Kutta approximation of inhomogeneous parabolic equations}, volume = {102}, journal = {Numer. Math.}, doi = {http://dx.doi.org/10.1007/s00211-005-0624-3}, pages = {277 -- 291}, year = {2005}, language = {en} } @article{LopezFernandezLubichSchaedle2008, author = {Lopez-Fernandez, Maria and Lubich, Christian and Sch{\"a}dle, Achim}, title = {Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory}, volume = {30}, journal = {SIAM J. Sci. Comput.}, number = {2}, doi = {/10.1137/060674168}, pages = {1015 -- 1037}, year = {2008}, language = {en} } @article{LopezFernandezPalenciaSchaedle2006, author = {Lopez-Fernandez, Maria and Palencia, Cesar and Sch{\"a}dle, Achim}, title = {A spectral order method for inverting sectorial Laplace transforms}, volume = {44}, journal = {SIAM J. Numer. Anal.}, number = {3}, doi = {/10.1137/050629653}, pages = {1332 -- 1350}, year = {2006}, language = {en} } @article{KalkbrennerHakansonSchaedleetal.2005, author = {Kalkbrenner, T. and Hakanson, U. and Sch{\"a}dle, Achim and Burger, Sven and Henkel, C. and Sandoghdar, V.}, title = {Optical microscopy using the spectral modifications of a nano-antenna}, volume = {95}, journal = {Phys. Rev. Lett.}, doi = {10.1103/PhysRevLett.95.200801}, pages = {200801}, year = {2005}, language = {en} } @inproceedings{SchaedleZschiedrich2007, author = {Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {Additive Schwarz method for scattering problems using the PML method at interfaces}, volume = {55}, booktitle = {Domain Decomposition Methods in Science and Engineering XVI}, editor = {Widlund, Olof and Keyes, David}, publisher = {Springer}, pages = {205 -- 212}, year = {2007}, language = {en} } @article{SchaedleZschiedrichBurgeretal.2007, author = {Sch{\"a}dle, Achim and Zschiedrich, Lin and Burger, Sven and Klose, Roland and Schmidt, Frank}, title = {Domain Decomposition Method for Maxwell's Equations}, volume = {226}, journal = {J. Comput. Phys.}, doi = {10.1016/j.jcp.2007.04.017}, pages = {477 -- 493}, year = {2007}, language = {en} } @article{SchmidtHohageKloseetal.2008, author = {Schmidt, Frank and Hohage, Thorsten and Klose, Roland and Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {Pole condition: A numerical method for Helmholtz-type scattering problems with inhomogeneous exterior domain}, volume = {218}, journal = {J. Comput. Appl. Math.}, number = {1}, doi = {10.1016/j.cam.2007.04.046}, pages = {61 -- 69}, year = {2008}, language = {en} } @article{RuprechtSchaedleSchmidtetal.2008, author = {Ruprecht, Daniel and Sch{\"a}dle, Achim and Schmidt, Frank and Zschiedrich, Lin}, title = {Transparent boundary conditions for time-dependent problems}, volume = {30}, journal = {SIAM J. Sci. Comput.}, number = {5}, doi = {10.1137/070692637}, pages = {2358 -- 2385}, year = {2008}, language = {en} } @article{SchaedleLopezFernandezLubich2006, author = {Sch{\"a}dle, Achim and L{\´o}pez-Fern{\´a}ndez, Mar{\´i}a and Lubich, Christian}, title = {Fast and oblivious convolution quadrature}, volume = {28}, journal = {SIAM J. Sci. Comput.}, number = {2}, doi = {http://dx.doi.org/10.1137/050623139}, pages = {421 -- 438}, year = {2006}, language = {en} } @inproceedings{SchaedleRuprechtSchmidt2007, author = {Sch{\"a}dle, Achim and Ruprecht, Daniel and Schmidt, Frank}, title = {Transparent boundary conditions - the pole condition approach}, booktitle = {Proceedings of Waves 2007}, editor = {et.al., N. Biggs}, pages = {301 -- 303}, year = {2007}, language = {en} } @inproceedings{SchmidtBurgerSchaedleetal.2005, author = {Schmidt, Frank and Burger, Sven and Sch{\"a}dle, Achim and Klose, Roland}, title = {Finite element methods for optical device design}, booktitle = {Proceedings of the 5th International Conference on Numerical Simulation of Optoelectronic devices}, doi = {10.1109/NUSOD.2005.1518133}, pages = {57 -- 58}, year = {2005}, language = {en} } @inproceedings{ZschiedrichBurgerKloseetal.2005, author = {Zschiedrich, Lin and Burger, Sven and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {JCMmode: an adaptive finite element solver for the computation of leaky modes}, volume = {5728}, booktitle = {Integrated Optics: Devices, Materials, and Technologies IX}, editor = {Sidorin, Y. and W{\"a}chter, Christoph}, publisher = {Proc. SPIE}, doi = {10.1117/12.590372}, pages = {192 -- 202}, year = {2005}, language = {en} } @inproceedings{ZschiedrichBurgerSchaedleetal.2008, author = {Zschiedrich, Lin and Burger, Sven and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {A rigorous finite-element domain decomposition method for electromagnetic near field simulations}, volume = {6924}, booktitle = {Optical Microlithography XXI}, editor = {Levinson, Harry and Dusa, Mircea}, publisher = {SPIE}, doi = {10.1117/12.771989}, year = {2008}, language = {en} } @inproceedings{ZschiedrichBurgerSchaedleetal.2005, author = {Zschiedrich, Lin and Burger, Sven and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {Domain decomposition method for electromagnetic scattering problems: application to EUV lithography}, booktitle = {Proceedings of the 5th International Conference on Numerical Simulation of Optoelectronic devices}, doi = {10.1109/NUSOD.2005.1518132}, pages = {55 -- 56}, year = {2005}, language = {en} } @article{ZschiedrichKloseSchaedleetal.2006, author = {Zschiedrich, Lin and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D}, volume = {188}, journal = {J. Comput. Appl. Math.}, doi = {10.1016/j.cam.2005.03.047}, pages = {12 -- 32}, year = {2006}, language = {en} } @misc{BurgerKloseSchaedleetal.2003, author = {Burger, Sven and Klose, Roland and Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {HelmPole - A finite element solver for scattering problems on unbounded domains: Implementation based on PML}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7609}, number = {03-38}, year = {2003}, abstract = {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.}, language = {en} } @misc{ZschiedrichKloseSchaedleetal.2003, author = {Zschiedrich, Lin and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {A new Finite Element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D}, doi = {10.1016/j.cam.2005.03.047}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7662}, number = {03-44}, year = {2003}, abstract = {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.}, language = {en} } @misc{SchaedleLopezFernandezLubich2005, author = {Sch{\"a}dle, Achim and Lopez-Fernandez, Maria and Lubich, Christian}, title = {Fast and oblivious convolution quadrature}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8437}, number = {05-09}, year = {2005}, abstract = {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.}, language = {en} } @misc{LopezFernandezLubichPalenciaetal.2005, author = {Lopez-Fernandez, Maria and Lubich, Christian and Palencia, Cesar and Sch{\"a}dle, Achim}, title = {Fast Runge-Kutta approximation of inhomogeneous parabolic equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8443}, number = {05-10}, year = {2005}, abstract = {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.}, language = {en} } @article{AntoineArnoldBesseetal.2008, author = {Antoine, Xavier and Arnold, Anton and Besse, Christophe and Ehrhardt, Matthias and Sch{\"a}dle, Achim}, title = {A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schr{\"o}dinger Equations}, volume = {4}, journal = {Commun. Comput. Phys.}, pages = {729 -- 796}, year = {2008}, language = {en} } @inproceedings{BurgerZschiedrichKloseetal.2005, author = {Burger, Sven and Zschiedrich, Lin and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank and Enkrich, Christian and Linden, S. and Wegener, M. and Soukoulis, C.}, title = {Numerical Investigation of Light Scattering off Split-Ring Resonators}, volume = {5955}, booktitle = {Metamaterials}, editor = {Szoplik, T. and {\"O}zbay, E. and Soukoulis, C. and Zheludev, N.}, publisher = {Proc. SPIE}, doi = {10.1117/12.622184}, pages = {18 -- 26}, year = {2005}, language = {en} } @inproceedings{BurgerZschiedrichSchaedleetal.2006, author = {Burger, Sven and Zschiedrich, Lin and Sch{\"a}dle, Achim and Klose, Roland and Kettner, Benjamin and Pomplun, Jan and Pollok, T. and Schmidt, Frank}, title = {Advanced FEM Analysis of Nano-Optical Devices}, booktitle = {Proc. Symposium on Photonics Technologies for the 7th Framework Programme}, year = {2006}, language = {en} } @inproceedings{BurgerSchaedleZschiedrichetal.2006, author = {Burger, Sven and Sch{\"a}dle, Achim and Zschiedrich, Lin and Schmidt, Frank}, title = {Advanced FEM Analysis of Nano-Optical Devices}, volume = {6195}, booktitle = {Nanophotonics}, editor = {Andrews, David L. and Nunzi, Jean-Michel and Ostendorf, Andreas}, publisher = {Proc. SPIE}, doi = {10.1117/12.662942}, pages = {72 -- 82}, year = {2006}, language = {en} } @inproceedings{BurgerKloseMaerzetal.2004, author = {Burger, Sven and Klose, Roland and M{\"a}rz, Reinhard and Sch{\"a}dle, Achim and Schmidt, Frank and Zschiedrich, Lin}, title = {Efficient Finite Element Methods for the Design of Microoptical Components}, booktitle = {Proc. Microoptics Conf. 2004}, year = {2004}, language = {en} } @inproceedings{BurgerKloseSchaedleetal.2005, author = {Burger, Sven and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank and Zschiedrich, Lin}, title = {FEM modelling of 3D photonic crystals and photonic crystal waveguides}, volume = {5728}, booktitle = {Integrated Optics : Devices, Materials and Technologies IX}, editor = {Sidorin, Y. and W{\"a}chter, Christoph}, publisher = {Proc. SPIE}, doi = {10.1117/12.585895}, pages = {164 -- 173}, year = {2005}, language = {en} } @incollection{BurgerKloseSchaedleetal.2006, author = {Burger, Sven and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank and Zschiedrich, Lin}, title = {Adaptive FEM solver for the computation of electromagnetic eigenmodes in 3D photonic crystal structures}, booktitle = {Scientific Computing in Electrical Engineering}, publisher = {Springer Verlag}, doi = {10.1007/978-3-540-32862-9_24}, pages = {169 -- 175}, year = {2006}, language = {en} } @article{RuprechtSchaedleSchmidt2013, author = {Ruprecht, Daniel and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {Transparent boundary conditions based on the pole condition for time-dependent, two-dimensional problems}, volume = {29}, journal = {Numerical Methods for Partial Differential Equations}, number = {4}, doi = {10.1002/num.21759}, pages = {1367 -- 1390}, year = {2013}, language = {en} } @misc{AntoineArnoldBesseetal.2007, author = {Antoine, Xavier and Arnold, Anton and Besse, Christophe and Ehrhardt, Matthias and Sch{\"a}dle, Achim}, title = {A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schr{\"o}dinger Equations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10332}, number = {07-34}, year = {2007}, abstract = {In this review article we discuss different techniques to solve numerically the time-dependent Schr{\"o}dinger 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.}, language = {en} } @misc{RuprechtSchaedleSchmidtetal.2007, author = {Ruprecht, Daniel and Sch{\"a}dle, Achim and Schmidt, Frank and Zschiedrich, Lin}, title = {Transparent boundary conditons for time-dependent problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9558}, number = {07-12}, year = {2007}, abstract = {A new approach to derive transparent boundary conditions (TBCs) for wave, Schr{\"o}dinger, 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 M{\"o}bius 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.}, language = {en} }