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Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 13-37. |
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Adaptive Hybridized Interior Penalty Discontinuous Galerkin Methods for H(curl)-Elliptic Problems C. Carstensen 1, R. H. W. Hoppe 2*, N. Sharma 3, T. Warburton 4 1 Department of Mathematics, Humboldt Universitat zu Berlin, D-10099 Berlin, Germany; and Department of Computer Science Engineering, Yonsei University, Seoul 120-749, Korea.2 Department of Mathematics, University of Houston, Houston TX 77204-3008, USA; and Institute of Mathematics, University of Augsburg, D-86159 Augsburg, Germany. 3 Department of Mathematics, University of Houston, Houston TX 77204-3008, USA. 4 CAAM, Rice University, Houston, TX 77005-1892, USA. Received 21 April 2010; Accepted (in revised version) 5 October 2010 Abstract We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin (IPDG-H) method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations. The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain. It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method. The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals. Within a unified framework for adaptive finite element methods, we prove the reliability of the estimator up to a consistency error. The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D. AMS subject classifications: 65N30, 65N50, 78M10Key words: Adaptive hybridized Interior Penalty Discontinuous Galerkin method, a posteriori error analysis, H(curl)-elliptic boundary value problems, semi-discrete eddy currents equations. *Corresponding author. Email: cc@math.hu-berlin.de (C. Carstensen), rohop@math.uh.edu (R. H. W. Hoppe), nsharma@math.uh.edu (N. Sharma), timwar@riceu.edu (T. Warburton) |