Optimal discretization-based adaptive finite element analysis for 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. Consequently, a major research issue in adaptive finite element analysis is the feedback control system used to guide the adaption; essentially, one needs to resolve which error data to feed back after each iteration, and how to use it to initialize the next adaptive step. Currently, there exists substantial evidence suggesting that the optimality of a finite element discretization plays a significant role in the accuracy of computed solutions at given levels of problem refinement. Therefore, in order to exploit the potential benefits in adaptive finite element methods, the characterization of optimal finite element discretizations has been investigated extensively. However, valid criteria for characterizing optimal finite element discretizations for a sufficiently wide range of problem applications have not been reported. A theoretical formulation for the numerical study of optimal finite element solutions to partial differential equations of macroscopic electromagnetics is presented. 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. The optimal characteristics of approximate finite element solutions, as predicted by the theory and observed numerically, have been employed to develop new optimal discretization-based feedback refinement criteria for use with advanced strategy adaption models in finite element electromagnetics. Numerical tests indicate that they are effective and economical for efficiently and reliably guiding practical h-, p- and hp-type adaption models towards accurate solutions.In addition, a series of important benchmark adaption problems are introduced to examine the validity of the theoretical concepts and the practical value of the new refinement criteria. Moreover, many of the computational and theoretical difficulties inherent in the currently available characterizations of optimal finite element discretizations are explained and illustrated with numerical results computed for the same benchmark problems