Submitted

Quasi-optimal convergence of adaptive edge finite element methods for three dimensional indefinite time-harmonic Maxwell's equations

Liuqiang Zhong, Long Chen, Shi Shu, Gabriel Wittum, and Jinchao Xu

To be submitted

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ABSTRACT:

We consider the simplest and most standard Adaptive Edge Finite
Element Method (AEFEM), with any order N\'{e}d\'{e}lec edge finite
element, for the 3D indefinite time-harmonic Maxwell's equations. As
is customary in practice, AEFEM marks exclusively according to the
error estimator without special treatment of oscillation and
performs a minimal element refinement without the interior node
property. We prove that the AEFEM is a contraction, for the sum of
the energy error and the scaled error estimator, between two
consecutive adaptive loops. This geometric decay is instrumental to
derive the optimal cardinality of the AEFEM. We show that the AEFEM
yields a decay rate of the energy error plus oscillation in terms of
the number of degrees of freedom as dictated by the best
approximation for this combined nonlinear quantity. Numerical
experiments are carried out to support the theoretic results.