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

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.