Computational Electromagnetism
Authors: Hiptmair Ralf , Hoppe Ronald H.W. , Langer Ulrich 
Published: December 2009

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 [2] arose), and the “LMS Durham Symposium on Computational methods for Wave Propagation in Direct Scattering”, Durham, England, July 15–25, 2002 (see [1]).
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:
We would like to add our personal impression that two families of methods have been received with particular interest during the workshop: