This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete di↵erential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of di↵erential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete di↵erential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and di culties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

In this paper we are concerned with the efficient solution of discrete variational problems related to the bilinear form (curl ·, curl ·) L 2 (Ω) + (·, ·) L 2 (Ω) defined on H 0 (curl; Ω). This is a core task in the time-domain simulation of electromagnetic fields, if implicit timestepping is employed. We rely on Nédélec's H (curl; Ω)-conforming finite elements (edge elements) to discretize the problem.We construct a multigrid method for the fast iterative solution of the resulting linear system of equations. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the curl operator, Helmholtz decompositions are the key to the design of the algorithm: N (curl) and its complement N (curl) ⊥ require separate treatment. Both can be tackled by nodal multilevel decompositions, where for the former the splitting is set in the space of discrete scalar potentials.Under certain assumptions on the computational domain and the material functions, a rigorous proof of the asymptotic optimality of the multigrid method can be given, which shows that convergence does not deteriorate on very fine grids. The results of numerical experiments confirm the practical efficiency of the method. Key words. Maxwell's equations, Nédélec's finite elements, edge elements, multilevel methods, multigrid AMS subject classifications. 65N55, 65N30, 65P05, 35Q60 PII. S00361429973262031. Introduction. One of the principal tasks of scientific computing is the efficient numerical simulation of electromagnetic fields. Their behavior is governed by Maxwell's equations [42], a system of first-order partial differential equations. We consider the scattering of electromagnetic waves in a bounded cavity Ω ⊂ R 3 , filled with a linear isotropic material. In this case Maxwell's equations lead to the following system of partial differential equations for the electric field E = E(x, t) and the magnetic field H = H(x, t) (see, e.g., [48]):

Summary. Methods based on boundary integral equations are widely used in the numerical simulation of electromagnetic scattering in the frequency domain. This article examines a particular class of these methods, namely the Galerkin boundary element approach, from a theoretical point of view. Emphasis is put on the fundamental differences between acoustic and electromagnetic scattering. The derivation of various boundary integral equations is presented, properties of their discretized counterparts are discussed, and a-priori convergence estimates for the boundary element solutions are rigorously established.