@misc{YevickFrieseSchmidt, author = {Yevick, David and Friese, Tilmann and Schmidt, Frank}, title = {A Comparison of Transparent Boundary Conditions for the Fresnel Equation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5738}, number = {00-05}, abstract = {We establish the relationship between the transparent boundary condition (BPP) of Baskakov and Popov [Wave Motion 14 (1991) 121-128] and Pakpadakis et. al. [J. Acoust. Soc. Am. 92 (1992) 2030-2038] and a second boundary condition (SDY) introduced by Schmidt and Deuflhard [Comp. Math. Appl. 29 (1995) 53-76] and Schmidt and Yevick [J. Compu. Phys. 134 (1997) 96-107], that is explicitly tailored to the form of the underlying numerical propagation scheme. Our analysis demonstrates that if the domain is first discretized in the propagation direction, the SDY expression can be obtained by applying the exact sequence of steps used to derive the BPP procedure. The BPP method is thus an approximate realization of the computationally far simpler and unconditionally stable SDY boundary condition.}, language = {en} } @misc{Friese, author = {Friese, Tilmann}, title = {MATLAB-Programme zur L{\"o}sung von Eigenwertproblemen mit einem adaptiven, nichtlinearen Mehrgitter-Verfahren.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5438}, number = {TR-97-01}, abstract = {Der Report beschreibt ein {\sc Matlab}--Programmpaket zur Bestimmung von Eigenl{\"o}sungen der skalaren, komplexen Helmholtzgleichung auf einem zweidimensionalen Gebiet. Die Programme erm{\"o}glichen die Berechnung der \$q\$ Eigenwerte mit kleinstem Realteil und der korrespondierenden Eigenfunktionen {\"u}ber adaptiv verfeinerten Gittern sowie die grafische Darstellung der L{\"o}sungen. Die bei der Diskretisierung entstehenden algebraischen Eigenwertprobleme werden mit einem nichtlinearen Mehrgitter--Verfahren gel{\"o}st.}, language = {de} } @phdthesis{Friese, author = {Friese, Tilmann}, title = {Eine Mehrgitter-Methode zur L{\"o}sung des Eigenwertproblems der komplexen Helmholtzgleichung}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:188-1999000133}, language = {en} } @misc{DeuflhardFrieseSchmidtetal., author = {Deuflhard, Peter and Friese, Tilmann and Schmidt, Frank and M{\"a}rz, Reinhard and Nolting, Hans-Peter}, title = {Effiziente Eigenmodenberechnung f{\"u}r den Entwurf integriert-optischer Chips}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2131}, number = {SC-96-02}, abstract = {{\bf Efficient eigenmode computation for the design of integrated optical chips.}The paper deals with adaptive multigrid methods for 2D Helmholtz eigenvalue problems arising in the design of integrated optical chips. Typical features of the technological problem are its geometric complexity, its multiscale structure, the possible occurrence of eigenvalue clusters, and the necessity of quite stringent required relative error tolerances. For reasons of sheer computational complexity, multigrid methods must be used to solve the discretized eigenvalue problems and adaptive grids must be automatically constructed to avoid an undesirable blow-up of the required number of nodes for these accuracies. In view of the problem specifications, an adaptive multigrid method based on Rayleigh quotient minimization, simultaneous eigenspace iteration, and conjugate gradient method as smoother is carefully selected. Its performance in the numerical simulation of a component of a rather recent optical chip (heterodyne receiver of HHI) is documented.}, language = {de} } @misc{SchmidtFrieseYevick, author = {Schmidt, Frank and Friese, Tilmann and Yevick, David}, title = {Transparent Boundary Conditions for Split-Step Pade Approximations of the One-Way Helmholtz Equation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4353}, number = {SC-99-46}, abstract = {In this paper, we generalize the nonlocal discrete transparent boundary condition introduced by Schmidt and Deuflhard {[}Comp. Math. Appl. 29 (1995) 53-76{]} and Schmidt and Yevick {[}J. Comput. Phys. 134 (1997) 96-107{]} to propagation methods based on arbitrary Pad\'e approximations to the two-dimensional one-way Helmholtz equation. Our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method.}, language = {en} } @misc{DeuflhardFrieseSchmidt, author = {Deuflhard, Peter and Friese, Tilmann and Schmidt, Frank}, title = {A Nonlinear Multigrid Eigenproblem Solver for the Complex Helmholtz Equation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3240}, number = {SC-97-55}, abstract = {The paper is motivated by the need for a fast robust adaptive multigrid method to solve complex Helmholtz eigenvalue problems arising from the design of optical chips. A nonlinear multigrid method is developed, which can be regarded as an extension of a previous adaptive Rayleigh quotient minimization method for selfadjoint Helmholtz eigenproblems. Since the complex Helmholtz operator is just a compact nonselfadjoint perturbation of a selfadjoint operator, linear algebra techniques like Schur decomposition can be extended from the finite dimensional case. The efficiency of the derived adaptive nonlinear multigrid method is illustrated by computations for a technologically relevant integrated optics component containing Multi Quantum Well Layers.}, language = {en} } @misc{Friese, author = {Friese, Tilmann}, title = {On the Computation of Eigenmodes of Integrated Optical Components}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3268}, number = {SC-97-57}, abstract = {The paper deals with the solution of the eigenvalue problem of the complex Helmholtz equation. We concentrate on multigrid methods for solving the algebraic eigenproblems arising from discretization with finite elements by using adaptive generated meshes. An illustrative numerical example, the simulation of a waveguide structure from integrated optics, is included.}, language = {en} } @misc{SchmidtFrieseZschiedrichetal., author = {Schmidt, Frank and Friese, Tilmann and Zschiedrich, Lin and Deuflhard, Peter}, title = {Adaptive Multigrid Methods for the Vectorial Maxwell Eigenvalue Problem for Optical Waveguide Design}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6228}, number = {00-54}, abstract = {This paper has been motivated by the need for a fast robust adaptive multigrid method to solve the vectorial Maxwell eigenvalue problem arising from the design of optical chips. Our nonlinear multigrid methods are based on a previous method for the scalar Helmholtz equation, which must be modified to cope with the null space of the Maxwell operator due to the divergence condition. We present two different approaches. First, we present a multigrid algorithm based on an edge element discretization of time-harmonic Maxwell's equations, including the divergence condition. Second, an explicit elimination of longitudinal magnetic components leads to a nodal discretization known to avoid discrete \emph{spurious modes} also and a vectorial eigenvalue problem, for which we present a multigrid solver. Numerical examples show that the edge element discretization clearly outperforms the nodal element approach.}, language = {en} }