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A convergent adaptive finite element method for photonic crystal fiber applications – 2009, MAFELAP 09, Brunel 

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In the last decades, mesh adaptivity has been widely used to improve the accuracy of numerical solutions to many scientific problems. Recently the question of convergence of adaptive methods has received intensive interest and a number of convergence results for the adaptive solution of boundary value problems have appeared.
We proved the convergence of an adaptive linear finite element algorithm for eigenvalue problems arising from photonic crystal fibres (PCFs). Our refinement procedure is based on two locally defined quantities, firstly a standard a posteriori error estimator and secondly a measure of the variability (or “oscillation”) of the computed eigenfunction. Our algorithm performs local refinement on all elements on which the minimum of these two local quantities is sufficiently large. We prove that the adaptive method converges provided the initial mesh is sufficiently fine. The latter condition, while absent for adaptive methods for linear symmetric elliptic boundary value problems, commonly appears for nonlinear problems and can be thought of as a manifestation of the nonlinearity of the eigenvalue problem.
Moreover, a number of numerical experiments concerning PCF problems shall be illustrated. In particular, we are interested in using our method to compute reliably band gaps for periodic media and trapped modes in PCFs with defects.