Published

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

Mathematics of Computation, 81(278), 623-642, 2012.

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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.