Convergence of adaptive hybridizable discontinuous Galerkin methods for second-order elliptic equations.

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Convergence of adaptive hybridizable discontinuous Galerkin methods for second-order elliptic equations.

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Zhang, Wujun

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2012-08

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Abstract

We present several a posteriori error estimators for the so-called hybridizable discontinuous Galerkin (HDG) methods, as well as an a posteriori error estimator for variabledegree, hybridized version of the Raviart-Thomas method on nonconforming meshes, for second-order elliptic equations. We show that the error estimators provide a reliable upper and lower bound for the true error of the flux in the L2-norm. Moreover, we establish the convergence and quasi-optimality of adaptive hybridizable discontinuous Galerkin (AHDG) methods. We prove that the so-called quasi-error, that is, the sum of an energy-like error and a suitably scaled error estimator, is a contraction between two consecutive loops. We also show that the AHDG methods achieve optimal rates of convergence.

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A posteriori error estimation and adaptivity

Discontinuous Galerkin method

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University of Minnesota Ph.D. dissertation. August 2012. Major: Mathematics. Advisor: Professor Bernardo Cockburn. 1 computer file (PDF); viii, 122 pages.

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Zhang, Wujun. (2012). Convergence of adaptive hybridizable discontinuous Galerkin methods for second-order elliptic equations.. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/139870.

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