A mixed method for axisymmetric div-curl systems

Abstract

Abstract. We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

Multigrid in a weighted space arising from axisymmetric electromagnetics

“…Since such details have been well explained by many authors [3,8,17], we shall not dwell on them here. It suffices to say that the 3D Maxwell system (consisting of six scalar equations for the six components of the electric and magnetic fields) decouples into two systems of three equations each, one called the azimuthal problem and the other the meridian problem.…”

“…Techniques to overcome the former difficulty are now well known, thanks to the papers [1,2,22], and we shall make extensive use of their ideas. Some important results to handle difficulties caused by the degenerate weight are available in [17,18,21]. However, we need to develop more tools in weighted spaces of vector-valued functions arising in our problem.…”

“…Simple finite elements for the azimuthal problem were analyzed in [21]. This was followed by an analysis of finite elements for the meridian problem [17]. Other studies on the performance of finite elements for axisymmetric Maxwell equations include [14,18,24].…”

Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier Singular Complement Method

“…If in addition the data are also invariant, then the problem can be further reduced to a single two-dimensional problem (cf. [5,4,24]). When this is not the case, one has to consider a series of two-dimensional problems, obtained by Fourier analysis.…”

“…On the other hand, edge element methods cannot be processed within the same framework. A mixed method, using edge elements conformal in a weighted H(curl)-type space to solve the static Maxwell equations with axisymmetric data, was described and analysed in [24]. In order to analyse a Fourier-Edge Element Method for timedependent Maxwell equations, one would have to combine this approach with that of [23], as well as Fourier analysis.…”