@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} }