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Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. (English) Zbl 0939.65136

Authors’ summary: 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 Nédélec’s \(H\)(\(\text{curl};\Omega)\)-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 esimators and the multigrid method is the separate treatment of the kernel of the curl operator and its orthogonal complement. Only 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 the 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.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N15 Error bounds for boundary value problems involving PDEs
78A50 Antennas, waveguides in optics and electromagnetic theory
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