Optimal discretizations in adaptive finite element electromagnetics (Q2770896)

One of the primary objectives of adaptive finite element analysis research is to determine how to effectively discretize a problem in order to obtain a sufficiently accurate solution efficiently. Therefore, the characterization of optimal finite element solution properties could have significant implications on the development of improved adaptive solver technologies. Ultimately, the analysis of optimally discretized systems, in order to learn about ideal solution characteristics, can lead to the design of better feedback refinement criteria for guiding practical adaptive solvers towards optimal solutions efficiently and reliably. A theoretical framework for the qualitative and numerical study of optimal finite element solutions to differential equations of macroscopic electromagnetics is presented in this study for one-, two- and three-dimensional systems. The formulation is based on variational aspects of optimal discretizations for Helmholtz systems that are closely related to the underlying stationarity principle used in computing finite element solutions to continuum problems. In addition, the theory is adequately general and appropriate for the study of a range of electromagnetics problems including static and time-harmonic phenomena. Moreover, finite element discretizations with arbitrary distributions of element sizes and degrees of approximating functions are assumed, so that the implications of the theory for practical \(h\)-, \(p\)-, \(hp\)- and \(r\)-type finite element adaption in multidimensional analyses may be examined.