Adaptively reducing boundary error in finite element analysis of unbounded electromagnetic wave problems

An adaptive finite element method is developed to solve two dimensional unbounded electromagnetic radiation and scattering problems. Though there has been considerable research in adaption in electromagnetics, to date, there have been few attempts to deal with adaptively reducing the error introduced by a boundary that artificially truncates the unbounded domain in finite element analysis. The technique proposed here adapts on this boundary error, as well as on the usual finite element discretization error. It combines three techniques: (1) p-adaptive hierarchal triangular finite elements, (2) wave-envelope elements, and (3) absorbing boundary conditions. Hierarchal finite elements allow the polynomial orders of the elements to be used to control the distribution of degrees of freedom: therefore, they make p-adaption possible, i.e., adaption by varying the element orders. This is more efficient than h-adaption, which requires actual re-meshing of the finite elements. The wave-envelope method uses a change of the dependent variable to remove the wave-like qualities of the solution and thereby permits the use of very large elements, i.e., elements much larger than a wavelength, in the external domain. An absorbing boundary condition is a boundary operator which approximately absorbs all the radiation incident on the boundary from within. In the new method, the scattering or radiating object itself, and its immediate surroundings, are meshed with hierarchal finite elements. Outside this, a thick layer of free space is meshed with hierarchal wave-envelope elements. The layer is thick enough such that when an absorbing boundary condition is imposed on its outer surface, there is very small reflection from it. Such a thick layer can be meshed with relatively few wave-envelope elements. The boundary error seen by the finite element region is, then, a function of how well the wave-envelope region is discretized. During p-adaption, increasing the order of the wave-envelope elements increases their ability to model the field accurately and, therefore, reduces the boundary error. Moreover, this reduction in the boundary error is selective: in directions of strong radiation, the error reduction is greater.