TY - GEN A1 - Beck, Rudolf A1 - Hiptmair, Ralf T1 - Multilevel Solution of the Time-Harmonic Maxwell's Equations Based on Edge Elements N2 - 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. T3 - ZIB-Report - SC-96-51 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2619 ER - TY - GEN A1 - Beck, Rudolf A1 - Deuflhard, Peter A1 - Hiptmair, Ralf A1 - Hoppe, Ronald H. W. A1 - Wohlmuth, Barbara T1 - Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations N2 - 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. T3 - ZIB-Report - SC-97-66 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3356 ER - TY - JOUR A1 - Beck, Rudolf A1 - Deuflhard, Peter A1 - Hiptmair, Ralf A1 - Hoppe, Ronald H. W. A1 - Wohlmuth, Barbara T1 - Adaptive multilevel methods for edge element discretizations of Maxwell’s equations JF - Surv. Math. Ind. Y1 - 1999 VL - 8 IS - 3-4 SP - 271 EP - 312 ER - TY - JOUR A1 - Hiptmair, Ralf A1 - Schädle, Achim T1 - Non-reflecting boundary conditions for Maxwell’s equations JF - Computing Y1 - 2003 UR - http://dx.doi.org/10.1007/s00607-003-0026-2 VL - 71 IS - 3 ER -