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.

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.
• Public URL: https://durham-repository.worktribe.com/output/1468231
• Related Public URLs: http://eprints.nottingham.ac.uk/1140/1/liang_et_al.pdf