A self-adaptive goal-oriented hp-finite element method with electromagnetic applications. Part II: Electrodynamics

We present the formulation, implementation, and applications of a self-adaptive, goal-oriented, hp-Finite Element (FE) Method for Electromagnetic (EM) problems. The algorithm delivers (without any user interaction) a sequence of optimal hp-grids. This sequence of grids minimizes the error in a prescribed quantity of interest with respect to the problem size, and it converges exponentially in terms of the relative error in a user-prescribed quantity of interest against the CPU time, including problems involving high material contrasts, boundary layers, and/or several singularities. The goal-oriented refinement strategy is an extension of a fully automatic, energy-norm based, hp-adaptive algorithm.We illustrate the efficiency of the method with 2D numerical simulations of Maxwell’s equations using both H1-conforming (continuous) elements and H(curl)-conforming (Nédélec edge) elements. Applications include alternate current (AC) resistivity logging instruments in a borehole environment with steel casing for the assessment of rock formation properties behind casing. Logging instruments, steel casing, and rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. For the presented challenging class of problems, the self-adaptive goal-oriented hp-FEM delivers results with 5–7 digits of accuracy in the quantity-of-interest.