A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity

Leszek Demkowicz; Jay Gopalakrishnan; Antti H. Niemi

doi:10.1016/j.apnum.2011.09.002

A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity

Leszek Demkowicz, Jay Gopalakrishnan, Antti H. Niemi

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Abstract

We continue our theoretical and numerical study on the Discontinuous Petrov-Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: ε=10 -11 for 1D and ε=10 -7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only. © 2011 IMACS. Published by Elsevier B.V. All rights reserved.

Original language	English (US)
Pages (from-to)	396-427
Number of pages	32
Journal	Applied Numerical Mathematics
Volume	62
Issue number	4
DOIs	https://doi.org/10.1016/j.apnum.2011.09.002
State	Published - Apr 2012
Externally published	Yes

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title = "A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity",

abstract = "We continue our theoretical and numerical study on the Discontinuous Petrov-Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: ε=10 -11 for 1D and ε=10 -7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only. {\textcopyright} 2011 IMACS. Published by Elsevier B.V. All rights reserved.",

author = "Leszek Demkowicz and Jay Gopalakrishnan and Niemi, {Antti H.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Demkowicz was supported in part by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615], and by a research contract with Boeing. Gopalakrishnan was supported in part by the National Science Foundation under grant DMS-0713833. Niemi was supported in part by KAUST. We thank Bob Moser and David Young for encouragement and stimulating discussions on the project. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2012",

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doi = "10.1016/j.apnum.2011.09.002",

language = "English (US)",

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journal = "Applied Numerical Mathematics",

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AU - Niemi, Antti H.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Demkowicz was supported in part by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615], and by a research contract with Boeing. Gopalakrishnan was supported in part by the National Science Foundation under grant DMS-0713833. Niemi was supported in part by KAUST. We thank Bob Moser and David Young for encouragement and stimulating discussions on the project. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - We continue our theoretical and numerical study on the Discontinuous Petrov-Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: ε=10 -11 for 1D and ε=10 -7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only. © 2011 IMACS. Published by Elsevier B.V. All rights reserved.

AB - We continue our theoretical and numerical study on the Discontinuous Petrov-Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: ε=10 -11 for 1D and ε=10 -7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only. © 2011 IMACS. Published by Elsevier B.V. All rights reserved.

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DO - 10.1016/j.apnum.2011.09.002

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VL - 62

SP - 396

EP - 427

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

IS - 4

ER -

A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity

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