Complementary Discrete Geometric h-Field Formulation for Wave Propagation Problems

An electromagnetic wave propagation problem can be formulated according to a pair of complementary formulations, called the e-formulation and the h-formulation. The two formulations are linked to each other by Maxwell’s curl equations, and, in the continuous setting, they are perfectly equivalent in describing the wave propagation phenomenon. However, this is not true in the discrete setting, where the two formulations, in general, give different solutions. In the past decades, complementary formulations were widely studied for static problems and eddy-current problems, where they were exploited as error estimators for adaptive refinement schemes. Moreover, the so-called bilateral energy bounds arise for some problems whether theoretically or at least numerically. However, to the best of our knowledge, little attention has been given to complementarity in the wave propagation problems. In this paper, we propose an adaptive refinement scheme using the constitutive error as an estimator, and then, we investigate the behavior in terms of bilateral energy bounds.