Abstract-Efficient functional derivative formulas suitable for optimal discretization based refinement criteria are developed for 3-D adaptive finite element analysis (FEA) with vector tetrahedra. Results for generalized vector Helmholtz systems are derived directly from first principles, and confirmed numerically through fundamental benchmark evaluations. Practical adaption applications are illustrated for selected FEA refinement models.Index Terms-Adaptive systems, electromagnetic analysis, error analysis, finite element methods.

-One of the major research issues in adaptive finite element analysis is the feedback control system used to guide the adaption. Essentially, one needs to resolve which error data to feedback after each iteration, and how to use it to initialize the next adaptive step. Variational aspects of optimal discretizations for scalar Poisson and Helmholtz systems are used to derive new refinement criteria for adaptive finite element solvers. They are shown to be effective and economical for h-,p-and hp-schemes.

-One of the most important problems of hybrid h-p adaption in finite element electromagnetics has been the accurate and efficient resolution of the singularities associated with sharp material edges and comers. One of the key obstacles has been the lack of objective standards by which to evaluate and compare adaptive control strategies. A set of optimal adaption benchmarks for the fundamental electromagnetic point and line singularity models is presented. The primary adaption procedures and control schemes are evaluated and compared. The absolute and relative performance of the competing approaches is discussed. started to emerge [5], [6], practical h-p adaptive strategies for electromagnetic FEA still remain out of reach. One important reason for this slow progress -aside from the inherent complexity of implementing and controlling h-p adaption -is the lack of objective benchmarks by which to measure the merits and flaws of adaptive strategies.One of the most important challenges for all types of adaption in FEA is the accurate and efficient resolution of the singularities associated with sharp material edges and corners [7]. The purpose of this contribution is to present a set of adaption benchmarks for these singularities, and illustrate their usefulness in the analysis and design of optimal h-p adaption strategies. I. INTRODUCTION II. ADAFTON BENCHMARKSCurrently, finite element analysis (FEA) is widely used in electromagnetic design -typically, FEA tools are used to computationally simulate and evaluate the performance of a new device design before building a prototype. Today, the state-of-the-art in FEA research lies in the development of adaptive solver technologies. In the future, it is believed that adaptive solvers will be able to reliably compute the performance of a proposed device to within the engineer's specified tolerances.Today. three basic adaption models are under study: htype; p-type; and combined h-and p-type (called h-p).Essentially, these models only differ in the techniques used to update the finite element discretization within the adaptive feedback loop (described below).A. Generate initial discretization. Repeat B. Solve finite element problem.C. Evaluate solution accuracy; if adequate STOP. D. Identify regions of inadequate discretization.E. Update finite element discretization. Until STOP.Simply stated, h-adaption adds elements to the mesh to improve a discretization; p-adaption increases element orders within the mesh to improve a discretization; and h-p adaption employs a combination of both procedures.While h-adaption has become increasingly popular in electromagnetic FEA research during the past ten years [ 11- [4], and more recently, effective p-adaption codes have The adaption analyses are based on the fundamental point-charge and line-current singularity models. The following benchmarks were computed for each model. (Orders 1 and 2). a) All nodes free to move with each adaptive step. b) Only new nodes free to move in an adaptive step. c) All new nodes set by element bi...

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