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In this paper we examine the well-know magneto-quasistatic eddy current model for the behaviour of low-frequency electromagnetic fields. We restrict ourselves to formulations in the frequency domain and linear materials, but admit rather general topological arrangements. The generic eddy current model allows two dual formulation, which may be dubbed E-based and H-based. We investigate so-called hybrid approaches that combine both formulation by means of coupling conditions across the boundaries of conducting regions. The resulting continuous and discrete variational formulations will be discussed. It is worthy to note that for this approach no difficulties arise from the topology of the conducting regions.

The computation of the resonant frequencies for open and closed cavities is not a trivial task: Multi-materials and sharp corners all give rise to highly singular eigenfunctions. However, an approach using hp-finite elements is well suited to such problems and, with the correct combination of h- and p-refinements, it yields the theoretically predicated exponential rates of convergence. In this paper, we present a novel approach to the solution of axisymmetric cavity problems which uses a hierarchic H1 and ${\vec H}(\hbox{curl})$ conforming finite element basis. A selection of numerical examples are included and these demonstrate that the exponential rates of convergence are achieved in practice., SAM Research Report, 2003-15

Many boundary integral equations for exterior Dirichlet- and Neumann boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary $\Gamma$ is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an $L^2 ({\Gamma})$-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes. This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods., SAM Research Report, 2003-08

We present a systematic study of how to take into account external excitation in the Eddy current model. Emphasis is put on mathematically sound variational formulations and on lumped parameter excitation through prescribed currents and voltages. We distinguish between local excitation at known contacts, known generator current distributions and non-local variants that rely on topological concepts. The latter case entails the violation of Faraday's law at so-called cuts and prevents us from reconstructing a meaningful electric field., SAM Research Report, 2003-07

Finite element approximations of eddy current problems that are entirely based on the magnetic field H are haunted by the need to enforce the algebraic constraint curlH = 0 in non-conducting regions. As an alternative to techniques employing combinatorial Seifert (cutting) surfaces in order to introduce a scalar magnetic potential, we propose mixed multi-field formulations, which enforce the constraint in the variational formulation. In light of the fact that the computation of cutting surfaces is expensive, the mixed finite element approximation is a viable option despite the increased number of unknowns.

We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderon projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods., SAM Research Report, 2001-05

We consider the electric field integral equation on the surface of polyhedral domains and its Galerkin-discretization by means of divergence-conforming boundary elements. With respect to a Hodge decomposition the continuous variational problem is shown to be coercive. However, this does not immediately carry over to the discrete setting, as discrete Hodge decompositions fail to possess essential regularity properties. Introducing an intermediate semidiscrete Hodge decomposition we can bridge the gap and come up with asymptotically optimal a-priori error estimates. Hitherto, those had been elusive, in particular for non-smooth boundaries., SAM Research Report, 2001-04

Multigrid method for H(div) in three dimensions. - In: Electronic transactions on numerical analysis. 6. 1997. S. 133-152

Nonconforming vector valued finite elements. - In: East-West Journal of numerical mathematics. 5. 1997. 3. S. 163-182

Multilevel preconditioned augmented lagrangian techniques for 2nd order mixed problems / R. Hiptmair ; T. Schiekofer ; B. Wohlmuth. - In: Computing. 57. 1996. S. 25-48