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- Algebraic multigrid (2)
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#### Institute

We present an algebraic multigrid preconditioner which uses only the graphs of system matrices. Some elementary coarsening rules are stated, from which an advancing front algorithm for the selection of coarse grid nodes is derived. This technique can be applied to linear Lagrange-type finite element discretizations; for higher-order elements an extension of the multigrid algorithm is provided. Both two- and three-dimensional second order elliptic problems can be handled. Numerical experiments show that the resulting convergence acceleration is comparable to classical geometric multigrid.

Our focus is on Maxwell's equations in the low frequency range; two specific applications we aim at are time-stepping schemes for eddy current computations and the stationary double-curl equation for time-harmonic fields. We assume that the computational domain is discretized by triangles or tetrahedrons; for the finite element approximation we choose N\'{e}d\'{e}lec's $H(curl)$-conforming edge elements of the lowest order. For the solution of the arising linear equation systems we devise an algebraic multigrid preconditioner based on a spatial component splitting of the field. Mesh coarsening takes place in an auxiliary subspace, which is constructed with the aid of a nodal vector basis. Within this subspace coarse grids are created by exploiting the matrix graphs. Additionally, we have to cope with the kernel of the $curl$-operator, which comprises a considerable part of the spectral modes on the grid. Fortunately, the kernel modes are accessible via a discrete Helmholtz decomposition of the fields; they are smoothed by additional algebraic multigrid cycles. Numerical experiments are included in order to assess the efficacy of the proposed algorithms.

KASKADE 3.0 User's Guide
(1995)

KASKADE 3.x was developed for the solution of partial differential equations in one, two, or three space dimensions. Its object-oriented implementation concept is based on the programming language C++$\,$.~Adaptive finite element techniques are employed to provide solution procedures of optimal computational complexity. This implies a posteriori error estimation, local mesh refinement and multilevel preconditioning. The program was designed both as a platform for further developments of adaptive multilevel codes and as a tool to tackle practical problems. Up to now we have implemented scalar problem types like stationary or transient heat conduction. The latter one is solved with the Rothe method, enabling adaptivity both in space and time. Some nonlinear phenomena like obstacle problems or two-phase Stefan problems are incorporated as well. Extensions to vector-valued functions and complex arithmetic are provided. This report helps to work with KASKADE Especially we \begin{itemize} \setlength{\parskip}{0ex} \item [{\bf --}] study a set of examples, \item [{\bf --}] explain how to define a user's problem and \item [{\bf --}] introduce a graphical user interface. \end{itemize} We are extending this guide continuously. The latest version is available by network.

KASKADE 3.0 was developed for the solution of partial differential equations in one, two, or three space dimensions. Its object-oriented implementation concept is based on the programming language C++$\,$.~Adaptive finite element techniques are employed to provide solution procedures of optimal computational complexity. This implies a posteriori error estimation, local mesh refinement and multilevel preconditioning. The program was designed both as a platform for further developments of adaptive multilevel codes and as a tool to tackle practical problems. Up to now we have implemented scalar problem types like stationary or transient heat conduction. The latter one is solved with the Rothe method, enabling adaptivity both in space and time. Some nonlinear phenomena like obstacle problems or two-phase Stefan problems are incorporated as well. Extensions to vector-valued functions and complex arithmetic are provided. %Such free boundary problems ... We have implemented several iterative solvers for both symmetric and unsymmetric systems together with multiplicative and additive multilevel preconditioners. Systems arising from the nonlinear problems can be solved with lately developed monotone multigrid methods. %An object-oriented concept was chosen for KASKADE~3.0, based on the programming %language C++$\,$. This should provide the desired extensibilty and clearly %reflect the structure of the code. %A direct sparse matrix solver (Harwell MA28) is included.

The focus of this paper is on the efficient solution of boundary value problems involving the double-- curl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling time--harmonic problems or in the context of eddy--current computations. Their discretization is based on on N\'ed\'elec's {\bf H(curl}; $\Omega$)--conforming edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the curl--operator and its orthogonal complement. Only on the latter we have proper ellipticity of the problem. Yet, exploiting the existence of computationally available discrete potentials for edge element spaces, we can switch to an elliptic problem in potential space to deal with nullspace of curl. Thus both cases become amenable to strategies of error estimation and multigrid solution developed for second order elliptic problems. The efficacy of the approach is confirmed by numerical experiments which cover several model problems and an application to waveguide simulation.

A widely used approach for the computation of time-harmonic electromagnetic fields is based on the well-known double-curl equation for either $\vec E$ or $\vec H$. An appealing choice for finite element discretizations are edge elements, the lowest order variant of a $H(curl)$-conforming basis. However, the large nullspace of the curl-operator gives rise to serious drawbacks. It comprises a considerable part of all spectral modes on the finite element grid, polluting the solution with non-physical contributions and causing the deterioration of standard iterative solvers. We tackle these problems by a nested multilevel algorithm. After every V-cycle in the $H(curl)$-conforming basis, the non-physical contributions are removed by a projection scheme. It requires the solution of Poisson's equation in the nullspace, which can be carried out efficiently by another multilevel iteration. The whole procedure yields convergence rates independent of the refinement level of the mesh. Numerical examples demonstrate the efficiency of the method.

Electromagnetic phased arrays for regional hyperthermia -- optimal frequency and antenna arrangement
(2000)

In this paper we investigate the effects of the three-dimensional arrangement of antennas and frequency on temperature distributions that can be achieved in regional hyperthermia using an electromagnetic phased array. We compare the results of power-based and temperature-based optimization. Thus we are able to explain the discrepancies between previous studies favouring more antenna rings on the one hand and more antennas per ring on the other hand. We analyze the sensitivity of the results with respect to changes in amplitudes and phases as well as patient position. This analysis can be used for different purposes. First, it provides additional criteria for selecting the optimal frequency. Second, it can be used for specifying the required phase and amplitude accuracy for a real phased array system. Furthermore, it may serve as a basis for technological developments in order to reduce both types of sensitivities described above.

Electromagnetic Phased Arrays for Regional Hyperthermia - Optimal Frequency and Antenna Arrangement
(2000)

After a short summary on therapy planning and the underlying technologies we discuss quantitative medicine by giving a short overview on medical image data, summarizing some applications of computer based treatment planning, and outlining requirements on medical planning systems. Then we continue with a description of our medical planning system {\sf HyperPlan}. It supports typical working steps in therapy planning, like data aquisition, segmentation, grid generation, numerical simulation and optimization, accompanying these with powerful visualization and interaction techniques.