Optimal discretizations in adaptive finite element electromagnetics

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 characterizati on of optimal finite element solution properties could have significant imp lications on the development of improved adaptive solver technologies. Ulti mately, the analysis of optimally discretized systems, in order to learn ab out ideal solution characteristics, can lead to the design of better feedba ck refinement criteria for guiding practical adaptive solvers towards optim al solutions efficiently and reliably. A theoretical framework for the qual itative and numerical study of optimal finite element solutions to differen tial equations of macroscopic electromagnetics is presented in this study f or one-, two- and three-dimensional systems. The formulation is based on va riational 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 electromagn etics problems including static and time-harmonic phenomena. Moreover, fini te element discretizations with arbitrary distributions of element sizes an d 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. Copyright (C) 2001 John Wiley & Sons, Ltd.