TY - JOUR A1 - Schütte, Christof A1 - Zumbusch, Gerhard A1 - Brinkmann, Ralf T1 - Dynamics of Erbium-doped Waveguide Lasers JF - preprint Y1 - 1995 UR - http://publications.imp.fu-berlin.de/104/ ER - TY - GEN A1 - Zumbusch, Gerhard T1 - Symmetric Hierarchical Polynomials and the Adaptive h-p-Version N2 - The $h$-$p$-version of finite-elements delivers a sub-exponential convergence in the energy norm. A step towards a full adaptive implementation is taken in the context of unstructured meshes of simplices with variable order $p$ in space. Both assumptions lead to desirable properties of shape functions like symmetry, $p$-hierarchy and simple coupling of elements. In a first step it is demonstrated that for standard polynomial vector spaces on simplices not all of these features can be obtained simultaneously. However, this is possible if these spaces are slightly extended or reduced. Thus a new class of polynomial shape functions is derived, which are especially well suited for three dimensional tetrahedra. The construction is completed by directly minimizing the condition numbers of the arising preconditioned local finite element matrices. The preconditioner is based on two-step domain decomposition techniques using a multigrid solver for the global linear problem $p=1$ and direct solvers for local higher order problems. Some numerical results concerning an adaptive (feedback) version of $h$-$p$ finite elements are presented. T3 - ZIB-Report - SC-95-18 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1841 ER - TY - GEN A1 - Schütte, Christof A1 - Dinand, Manfred A1 - Zumbusch, Gerhard A1 - Brinkmann, Ralf T1 - Dynamics of Erbium-doped Waveguide Lasers: Modelling, Reliable Simulation, and Comparison with Experiments N2 - A theoretical investigation of the dynamic properties of integrated optical Er--doped waveguide lasers is presented. It includes the construction of a physical model and of numerical techniques which allow reliable simulations of the dynamical behaviour of the laser signal depending on essential parameters of the laser device and on its external, time--dependent pump radiation. Therefore, a physical theory is developed which describes the propagation of light and its interaction with the active substrate in the laser cavity. This is realized in two steps. First, a {\em fundamental model} based on Maxwell's equations and on rate equations for the transitions in the active medium is constructed. Since this turns out to prohibit reliable simulations, it is, in a second step, reformulated via averaging in time and space which suppresses the fluctuations on the fastest time scales but represents them correctly. For this {\em reduced model} reliable and efficient simulation techniques using adaptive control schemes are designed and implemented. We apply the linear--implicit Euler discretization with extrapolation in time and a multilevel quadrature scheme in space. Finally, the model is justified in comparison with experimental observations in four cases of technological relevance. T3 - ZIB-Report - SC-95-19 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1856 ER - TY - GEN A1 - Zumbusch, Gerhard T1 - Adaptive h-p approximation procedures, graded meshes and anisotropic refinement for Numerical Quadrature N2 - A set of adaptive algorithms for quadrature on multi-dimensional polyhedral domains is presented. Several kinds of refinement are discussed, covering local improvement of quadrature order and splitting the domain into sub-domains, resulting in isotropic, graded or anisotropic grids. The algorithms are pure local heuristics using no a priori knowledge or tuning parameters. This approach was motivated by results from finite element theory for optimal approximation results. Numerical experiments show the optimality of pure local greedy-like algorithms for singularity-type functions typically occurring in finite element computations. T3 - ZIB-Report - SC-95-24 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1905 ER - TY - GEN A1 - Zumbusch, Gerhard T1 - Symmetric Hierarchical Polynomials for the h-p-Version of Finite Elements. N2 - Adaptive numerical methods using the $h$-$p$-version of finite elements require special kinds of shape functions. Desirable properties of them are symmetry, hierarchy and simple coupling. In a first step it is demonstrated that for standard polynomial vector spaces not all of these features can be obtained simultaneously. However, this is possible if these spaces are extended. Thus a new class of polynomial shape functions is derived, which is well-suited for the $p$- and $h$-$p$-version of finite elements on unstructured simplices. The construction is completed by minimizing the condition numbers of the arising finite element matrices. The new shape functions are compared with standard functions widely used in the literature. T3 - ZIB-Report - SC-93-32 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1275 ER - TY - GEN A1 - Zumbusch, Gerhard T1 - Visualizing Functions of the h-p-version of finite elements. N2 - Results from finite-element-calculations are usually visualized by colored surface- and contour-line-plots or polygonal patches or simply displaced lines and grids. In computer graphics however more advanced techniques like texture-mapping and NURBS are well established and there exist efficient algorithms and implementations. We show that these techniques are not only easy to use, but form a very natural and far more efficient approach for visualization of higher order finite-element's solutions like in $p$- and $h$-$p$-version. Texture-mapping is useful for displaying vector-valued data, too. T3 - ZIB-Report - TR-94-05 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5108 ER - TY - THES A1 - Zumbusch, Gerhard T1 - Simultanous h-p Adaption in Multilevel Finite Elements T3 - ZIB-Report - TR-95-14 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5298 ER -