A self-adaptive goal-oriented hp finite element method with electromagnetic applications. Part II: Electrodynamics D. Pardo a,b, * , L. Demkowicz a , C. Torres-Verdı ´n a,b , M. Paszynski a,1 a Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712, United States b Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712, United States Received 1 November 2005; accepted 30 October 2006 Abstract 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 H 1 -conforming (contin- uous) elements and H(curl)-conforming (Ne ´de ´lec edge) elements. Applications include alternate current (AC) resistivity logging instru- ments 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 pre- sented challenging class of problems, the self-adaptive goal-oriented hp-FEM delivers results with 5–7 digits of accuracy in the quantity- of-interest. Ó 2007 Elsevier B.V. All rights reserved. Keywords: hp-finite elements; Exponential convergence; Goal-oriented adaptivity; Computational electromagnetism; Maxwell’s equations; Through ca- sing resistivity tools (TCRT) 1. Introduction During the last decades, different algorithms have been designed and implemented to generate optimal grids in the solution of relevant engineering problems. Among those algorithms, a self-adaptive, energy-norm based, hp- Finite Element (FE) refinement strategy has been devel- oped at the Institute for Computational Engineering and Sciences (ICES) of The University of Texas at Austin. The strategy produces automatically a sequence of hp- meshes that delivers exponential convergence rates in terms of the energy-norm error against the number of unknowns (as well as the CPU time), independently of the number and type of singularities in the problem. Thus, it provides high accuracy approximations of solutions corresponding to a variety of engineering applications. Furthermore, the self-adaptive strategy is problem independent, and it can be applied to FE discretizations of H 1 -, H(curl)-, and H(div)-spaces, as well as to nonlinear problems (see [7,18] for details). The self-adaptive strategy iterates along the following steps. A given (coarse) conforming hp mesh is first globally refined in both h and p to yield a fine mesh, i.e. each element is broken into eight new elements, and the discretization order of approximation p is raised uniformly by one. 0045-7825/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2006.10.016 * Corresponding author. Address: Institute for Computational Engi- neering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712, United States. Tel.: +1 512 471 3312; fax: +1 512 471 8694. E-mail address: dzubiaur@gmail.com (D. Pardo). 1 Department of Computer Science, AGH University of Science and Technology, Cracow, Poland. www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. xxx (2007) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented hp finite element method ..., Comput. Methods Appl. Mech. Engrg. (2007), doi:10.1016/j.cma.2006.10.016