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 -