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

Demkowicz, Leszek; Gopalakrishnan, Jay; Niemi, Antti H.

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

Type

Article

Authors

Demkowicz, Leszek
Gopalakrishnan, Jay
Niemi, Antti H.

Date

2012-04

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.

Citation

Demkowicz L, Gopalakrishnan J, Niemi AH (2012) A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity. Applied Numerical Mathematics 62: 396–427. Available: http://dx.doi.org/10.1016/j.apnum.2011.09.002.

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.