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Adaptive finite element methods for photonic crystal fibers – 2007, Waves, Reading 

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We consider eigenvalue problems arising from the modeling of photonic crystal fibers (PCFs) and in particular we are interested in computing the first gaps in the bottom of the spectrum of periodic structures and in the trapped modes coming from the defects. In order to locate the gaps we split the Maxwell’s equations into the TE and TM modes. Then we apply the Bloch-Floquet transform obtaining two families of eigenvalue problems on a primitive cell P parameterized by the quasimomentum. In this talk we solve both families of problems using linear finite elements on triangular meshes and we propose a reliable and efficient a posteriori error estimate to drive mesh adaptivity. The a posteriori error estimator we use is based on residuals. This kind of estimate is quite common in finite element methods for partial differential equations, but there are relatively few results for eigenvalue problems. We are not aware so far of any a posteriori error estimates or mesh adaptivity for photonic eigenvalue problems.