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Adaptive discontinuous Galerkin methods for elliptic interface problems


Authors: Andrea Cangiani, Emmanuil H. Georgoulis and Younis A. Sabawi
Journal: Math. Comp. 87 (2018), 2675-2707
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3322
Published electronically: February 20, 2018
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Abstract: An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes, is considered. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. A residual-type a posteriori error estimator for this dG method is proposed and upper and lower bounds of the error in the respective dG-energy norm are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. The theory presented is complemented by a series of numerical experiments. The presented approach applies immediately to the case of curved domains with non-essential boundary conditions, too.


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Additional Information

Andrea Cangiani
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom
Email: Andrea.Cangiani@le.ac.uk

Emmanuil H. Georgoulis
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom; and Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou 15780, Greece
Email: Emmanuil.Georgoulis@le.ac.uk

Younis A. Sabawi
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom; and Department of Mathematics, Faculty of Science and Health, University of Koya, Kurdistan, Iraq; and Department of Mathematics Education, Faculty of Education, University of Ishik, Kurdistan, Iraq
Email: younis.sabawi@ishik.edu.iq

DOI: https://doi.org/10.1090/mcom/3322
Received by editor(s): September 16, 2016
Received by editor(s) in revised form: May 31, 2017
Published electronically: February 20, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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