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An a Posteriori Error Estimator for hp-Adaptive Discontinuous Galerkin Methods for Elliptic Eigenvalue Problems – 2011, FEMTEC 2011, Reno 

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In this talk we present a residual-based a posteriori error estimator for hp-adaptive discontinuous Galerkin (DG) methods for elliptic eigenvalue problems. In particular we use as a model problem the Laplace eigenvalue problem on bounded domains, with homogeneous Dirichlet boundary conditions. The same kind of error estimator can be easily extended to more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of our error estimator. The reliability ensures that, up to a constant and to asymptotic high order terms, the error estimator gives rise to an a posteriori error bound for both eigenvalues and eigenfunctions, on the other hand, the efficiency ensures that, up to a constant and to asymptotic high order terms, the true error bounds the error estimator. Together these two results ensures that the error estimator is linearly proportional to the true error, up to higher order terms. The ratio of the constants in the upper and lower bounds is independent of both the local mesh sizes and the local polynomial degrees. We apply our error estimator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples.