Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions

Zhu, Liang; Giani, Stefano; Houston, Paul; Schoetzau, Dominik

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 errors 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., & Schoetzau, D. Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions. Manuscript submitted for publication

• Journal Article Type: Article
• Deposit Date: Oct 2, 2009
• Peer Reviewed: Not Peer Reviewed
• Public URL: https://nottingham-repository.worktribe.com/output/1015029
• Additional Information: Preprint of an article submitted for consideration in Mathematical Models and Methods in Applied Sciences (M3AS) ©, 2009 [copyright World Scientific Publishing Company]http://www.worldscinet.com/m3as/m3as.shtml