Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations

Abstract

We consider the simplest and most standard Adaptive Edge FiniteElement Method (AEFEM), with arbitrary order Nédélec edge finiteelement, for three-dimensional indefinite time-harmonic Maxwellequations. We prove that the AEFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutiveadaptive loops provided the initial mesh is fine enough. Using thegeometric decay, we show that the AEFEM yields the best-possible decayrate of the error plus oscillation in terms of the number of degreesof freedom. The main technical difficulty is in the establishment of a quasi-orthogonality and a localized a posteriori error estimator.