A Posterior <i>h/p</i> Adaptive Refinement Algorithm for 3-D Goal-Oriented Electromagnetic Problems

Abstract

We propose a posterior h/p adaptive mesh refinement algorithm for 3-D goal-oriented electromagnetic wave modeling in complex media. This new algorithm allows both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> -version refinement—mesh size adjustment—and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -version refinement—polynomial order adjustment—simultaneously. To be specific, we use the adjoint system on the dual space of the forward solver; as a result, error generation and propagation can be evaluated with the system matrix and right-hand-side vector, even without the analytical solutions. In particular, the posterior error is explicitly indicated by postprocessing of original forward solver. Compared with conventional mesh adaptation strategies, such an algorithm is goal-oriented and auxiliary-system-free; as a result, it does not require extensive attention to entire computational domain and is complimentary in solving the adjoint state equation. Numerical examples demonstrate the superior performance of the proposed mesh refinement algorithms for real-world applications, compared to uniform refinement, nongoal-oriented adaptation, and other goal-oriented adaptation strategies.