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Convergence analysis of an hp finite element method for singularly perturbed transmission problems in smooth domains

Authors

  • Serge Nicaise,

    Corresponding author
    1. Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, F-59313 – Valenciennes Cedex 9, France
    • Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, F-59313 – Valenciennes Cedex 9, France
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  • Christos Xenophontos

    1. Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus
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Abstract

We consider a two-dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high and interface layers along the interface. Utilizing existing and newly derived regularity results for the exact solution, we prove the robustness of an hp finite element method for its approximation. Under the assumption of analytic input data, we show that the method converges at an “exponential” rate, provided the mesh and polynomial degree distribution are chosen appropriately. Numerical results illustrating our theoretical findings are also included. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

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