# Computational Electromagnetism

### Ralf Hiptmair

Eidgenössische Technische Hochschule, Zürich, Switzerland### Ronald H.W. Hoppe

Universität Augsburg, Germany### Ulrich Langer

Johannes Kepler Universität Linz, Austria

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## Abstract

The field of computational electromagnetism is dedicated to the design and analysis of numerical methods for the approximate solution of electromagnetic field problems. Since the exploitation of electromagnetic phenomena is one of the foundations of modern technology, computational electromagnetics is of tremendous industrial relevance: in a sense, it is peer to computational solid and fluid mechanics and huge research efforts are spent on developing and enhancing simulation methods and software for electromagnetic field computations. For a long time, computational electromagnetism remained a realm of engineering research with applied mathematics shunning the area. This was in stark contrast to elasticity and fluid mechanics, where mathematicians have been involved in the development of numerical methods from the very beginning. Maybe, the blame has to be laid on the incorrect belief of mathematicians who thought that the laws governing the behavior of electromagnetic fields basically boil down to well understood second-order elliptic problems. Fortunately, the past fifteen years have seen a real surge of mathematical research activities in the area of computational electromagnetism. This resulted in insights that have begun to have a big impact on the numerical methods used in engineering and industrial environments. A prominent example is the explanation of so-called spurious solutions that can arise when using continuous ``nodal'' finite elements for the discretization of certain electromagnetic boundary value or eigenvalue problems, respectively. Another example is the appreciation of so-called edge finite elements and the construction of multilevel iterative solvers for the low-frequency setting. Meanwhile, computational electromagnetism can claim to be a major area of numerical mathematics and scientific computing in its own right. This prompted us to ask the Mathematisches Forschungsinstitut Oberwolfach to host a one week workshop on computational electromagnetism, the first of its kind. Reflecting the growing importance of the subject, this workshop has been one of a series of events dedicated to mathematical issues in the computation of electromagnetic fields. We would like to mention, the NSF-CBMS Regional Conference in the Mathematical Sciences about ``Numerical Methods in Forward and Inverse Electromagnetic Scattering'', held in Golden, CO, June 3-7, 2002 (from which the book \cite{MON03} arose), and the ``LMS Durham Symposium on Computational methods for Wave Propagation in Direct Scattering'', Durham, England, July 15-25, 2002 (see \cite{ADD03}). This Oberwolfach workshop brought together some 50 experts in computational electromagnetism. The majority of the participants were applied mathematicians, but a sizable number of people with a background in engineering also attended, as appropriate for a field with close ties to engineering and the applied sciences. Nevertheless, the workshop had a clear mathematical focus, emphasizing rigorous theory, principles and ideas. Throughout, the presentations matched these expectations. A total of 29 presentations were given, of which ten were survey lectures offering broader treatment of a particular subject. As is typical of an event that targets a specific area of application, it arose that a broad range of mathematical issues and techniques was addressed. Although it will certainly not do justice to many presentations, we will try categorize the talks as follows: \begin{itemize} \item \textbf{Mathematical modelling.} This subject did not play a central role, because most presentations took the model equations for granted. Modelling for practical engineering calculations was described by O. B\'{i}r\'{o} in his survey talk about \emph{Practical Aspects of FEM in Electromagnetics}, p.~~\pageref{Biro}, and by M. Clemens when speaking on \emph{Formulations and Efficient Numerical Solution Techniques for Transient 3D Magneto-and Electro-Quasistatic Field Problems}, p.~~\pageref{Clemens}. Homogenization was addressed in the presentation by A. Bendali about \emph{Two Scale Asymptotic Expansion for the Scattering of a TM-Electromagnetic Wave by a Rough Surface and Applications}, p.~~\pageref{Bendali}. \item \textbf{Spatial discretizations.} This turned out to be one of the core subjects of the workshop. The survey lectures of D. Boffi about \emph{Theoretical Aspects of Edge Finite Elements}, p.~~\pageref{Boffi} and I. Perugia on \emph{Discontinuous Galerkin Methods for Maxwell's Equations}, p.~~\pageref{Perugia}, addressed the topic. Particular issues were discussed by S. Christiansen in his talk about the \emph{Div-Curl Lemma for Edge Elements}, p.~~\pageref{Christiansen}, and by J. Pasciak about \emph{The Approximation of the Maxwell Eigenvalue Problem using a Least-Squares Method}, p.~~\pageref{Pasciak}. M. Kaltenbacher gave an account of observations concerning finite element schemes in his presentation on \emph{Nodal and Edge Finite Element Discretization of Maxwell's Equations}, p.~~\pageref{Kaltenbacher}. Several presentations were devoted to higher order spatial discretization: the survey lecture of M. Ainsworth gave an account of the \emph{Dispersive Properties of High Order N\'ed\'elec/Edge Elements for Maxwell's Equations}, p.~~\pageref{Ainsworth}, L. Demkowicz spoke about \emph{$H^1,{\boldsymbol H}({\rm curl})$ and ${\boldsymbol H}({\rm div})$-Conforming Projection-Based Interpolation in Three Dimensions}, p.~~\pageref{Demkowicz}, and P. Ledger about \emph{Computation of Maxwell Eigenvalues with Exponential Rates of Convergence}. \item \textbf{Timestepping.} There was only one contribution dealing with temporal discretization, namely the talk by T. Driscoll on \emph{High-Order Time Stepping Methods for Electromagnetics}, p.~~\pageref{Driscoll}. \item \textbf{Regularity of solutions.} Here one of the pioneers in the field, M. Dauge, gave a survey talk about \emph{Singularities of Electromagnetic Fields in the Eddy Current Limit}, p.~~\pageref{Dauge}. \item \textbf{Integral equation methods}. Boundary element methods in the frequency domain were treated by S. Kurz in his talk on \emph{A New View on Collocation}, p.~~\pageref{Kurz}. Conversely, time-domain integral equation methods were examined in the survey lecture by E. Michielssen on \emph{Fast Time Domain Integral Equation Solvers}, p.~~\pageref{Michielssen}, and P. Davies in her contribution on \emph{Convergence of Collocation Methods for Time Domain Boundary Integral Equations}, p.~~\pageref{Davies}. S. B\"orm talked about \emph{${\mathcal H}^2$-Matrices with Adaptive Cluster Bases Applied to an Eddy Current Problem}, p.~~\pageref{Boerm}, and presented a fast summation method for discrete frequency-domain integral equations. \item \textbf{Electromagnetic Scattering.} This topic was treated by R. Kress in his survey lecture on \emph{Inverse Obstacle Scattering for Time-Harmonic Electromagnetic Waves}, p.~~\pageref{Kress}. Also the talk by A. Bendali on \emph{Two Scale Asymptotic Expansion for the Scattering of a TM-Electromagnetic Wave by a Rough Surface and Applications}, p.~~\pageref{Bendali}, addressed a particular scattering problem. \item \textbf{Absorbing boundary conditions.} A special incarnation of these was examined in the survey talk by F. Teixeira on \emph{Perfectly Matched Layers}, p.~~\pageref{Teixeira}. Details of a PML approach were studied by Z. Chen in his talk about \emph{An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems}, p.~~\pageref{Chen}. Other techniques were outlined by M. Grote (\emph{Nonreflecting Boundary Conditions for Computational Electromagnetics}, p.~~\pageref{Grote}) and F. Schmidt (\emph{Pole Condition: A New Approach to Solve Scattering Problems}, p.~~\pageref{Schmidt}). \item \textbf{Topological issues}. These were discussed in the talks of R. Kotiuga (\emph{The Hurewicz Map Distinguishes Intuitive vs. Computable Topological Aspects of Computational Electromagnetics}, p.~~\pageref{Kotiuga}) and F. Rapetti (\emph{Smith Normal Form as an Adequate Tool to Detect Mesh Defects as well as to Build Basis Fields for Domains with Loops and Holes}, p.~~\pageref{Rapetti}). \item \textbf{Fast solvers.} Several speakers discussed fast algorithms for the solution of linear systems of equations arising from discretized field equations: it was the subject of J. Sch\"oberl's survey lecture on \emph{Preconditioning for Maxwell Equations}, p.~~\pageref{Schoeberl}, and O. Sterz' talk on \emph{Adaptive Multigrid-Methods for the Solution of Time-Harmonic Eddy-Current Problems}, p.~~\pageref{Sterz}. The use of multigrid methods was discussed in the contributions by M. Clemens on \emph{Formulations and Efficient Numerical Solution Techniques for Transient 3D Magneto-and Electro-Quasistatic Field Problems}, p.`\pageref{Clemens}, and by M. Kaltenbacher on \emph{Nodal and Edge Finite Element Discretization of Maxwell's Equations}, p.~~\pageref{Kaltenbacher}. An enhancement for algebraic multigrid was proposed by P. Arbenz (\emph{Treatment of Nullspace in Maxwell Problem}, p.~~\pageref{Arbenz}). J. Zou dealt with domain decomposition methods in his contribution on \emph{Some New Inexact Uzawa Methods and Non-overlapping DD Preconditioners for Solving Maxwell's Equations in Non-homogeneous Media}, p.~~\pageref{Zou}. \item \textbf{Adaptive techniques.} Only one presentation, that of Z. Chen on \emph{An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems}, p.~~\pageref{Chen}, dealt with a special adaptive scheme. \item \textbf{Optimization.} This important subject reaches beyond the core of computational electromagnetism. An aspect was discussed in the talk by D. Luk\'{a}\v{s} on \emph{Computational Shape and Topology Optimization with Applications to 3--Dimensional Magnetostatics} p.~\pageref{Lukas}. \end{itemize} We would like to add our personal impression that two families of methods have been received with particular interest during the workshop: \begin{itemize} \item Time-domain integral equation methods, \item High-order spatial discretization schemes. \end{itemize} We are sure that the workshop will have made a substantial contribution to the progress of research in these and all other areas of computational electromagnetism.\bigskip \noindent\mbox{$$}\hfill\parbox{0.3\textwidth}{R. Hiptmair\\R.H.W. Hoppe\\U. Langer} \begin{thebibliography}{1} \bibitem{ADD03} {\sc M.~Ainsworth, P.~Davis, D.~Duncan, P.~Martin, and B.~Rynne}, eds., {\em Topics in Computational Wave Propagation. {D}irect and inverse Problems}, vol.~31 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, 2003. \bibitem{MON03} {\sc P.~Monk}, {\em Finite Element Methods for Maxwell's Equations}, Clarendon Press, Oxford, UK, 2003. \end{thebibliography}