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Convergent adaptive FEM for eigenvalue problems applied to photonic crystal fibres – 2007, NAMMAC, Bath 

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In this talk we shall present a convergent adaptive finite element method for eigenvalue problems arising from photonic crystal fibres (PCFs). The adaptive method we designed uses a marking strategy based on an a posteriori error estimator, together with a second marking strategy devoted to control the oscillations of computed eigenfunctions on the meshes. The convergence result holds under the condition that the starting mesh is fine enough for the particular problem. To keep the talk as simple as possible, we present the convergent result for a simple elliptic eigenvalue problem. Also 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.