Contraction property of adaptive hybridizable discontinuous galerkin methods

Bernardo Cockburn; Ricardo H. Nochetto; Wujun Zhang

doi:10.1090/mcom/3014

Contraction property of adaptive hybridizable discontinuous galerkin methods

Bernardo Cockburn, Ricardo H. Nochetto, Wujun Zhang

School of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	1113-1141
Number of pages	29
Journal	Mathematics of Computation
Volume	85
Issue number	299
DOIs	https://doi.org/10.1090/mcom/3014
State	Published - 2016

Bibliographical note

Funding 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.

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© 2015 American Mathematical Society.

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10.1090/mcom/3014

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abstract = "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.",

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AU - Cockburn, Bernardo

AU - Nochetto, Ricardo H.

AU - Zhang, Wujun

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PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

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Contraction property of adaptive hybridizable discontinuous galerkin methods

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