Energy Norm A-Posteriori Error Estimation for hp-Adaptive Discontinuous Galerkin Methods for Elliptic Problems in Three Dimensions

Zhu, L.; Giani, S.; Houston, P.; Schötzau, D.

doi:10.1142/s0218202511005052

Energy Norm A-Posteriori Error Estimation for hp-Adaptive Discontinuous Galerkin Methods for Elliptic Problems in Three Dimensions

Zhu, L.; Giani, S.; Houston, P.; Schötzau, D.

Authors

L. Zhu

Dr Stefano Giani stefano.giani@durham.ac.uk
Associate Professor

P. Houston

D. Schötzau

Abstract

We develop the energy norm a posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the error measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

Citation

Zhu, L., Giani, S., Houston, P., & Schötzau, D. (2011). Energy Norm A-Posteriori Error Estimation for hp-Adaptive Discontinuous Galerkin Methods for Elliptic Problems in Three Dimensions. Mathematical Models and Methods in Applied Sciences, 21(02), 267-306. https://doi.org/10.1142/s0218202511005052

Journal Article Type	Article
Publication Date	Feb 1, 2011
Deposit Date	Feb 12, 2013
Publicly Available Date	Apr 20, 2015
Journal	Mathematical Models and Methods in Applied Sciences
Print ISSN	0218-2025
Electronic ISSN	1793-6314
Publisher	World Scientific Publishing
Peer Reviewed	Peer Reviewed
Volume	21
Issue	02
Pages	267-306
DOI	https://doi.org/10.1142/s0218202511005052
Keywords	Discontinuous Galerkin methods, a posteriori error estimation, hp-adaptivity, Elliptic problems.
Related Public URLs	http://eprints.nottingham.ac.uk/1140/1/liang_et_al.pdf

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Copyright Statement
Electronic version of an article published as Mathematical Models and Methods in Applied Sciences, 21, 02, 2011, 267-306, doi: 10.1142/S0218202511005052 © copyright World Scientific Publishing Company http://www.worldscientific.com/worldscinet/m3as