We establish the contraction property between consecutive loops of adaptive hybridizable discontinuous Galerkin methods for the Poisson problem with homogeneous Dirichlet condition. The contractive quantity is the sum of the square of the L2-norm of the flux error, which is not even monotone, and a two-parameter scaled error estimator, which quantifies both the lack of H(div, Ω)-conformity and the deviation from a gradient of the approximate flux. A distinctive and novel feature of this analysis, which enables comparison between two nested meshes, is the lifting of trace residuals from inter-element boundaries to element interiors.
Bibliographical noteFunding Information:
The first author was partially supported by NSF Grant DMS-1115331 and by the Minnesota Supercomputing Institute. The second author was partially supported by NSF Grant DMS-1109325. The third author was partially supported by NSF Grants DMS-1115331 and DMS-1109325.