Skip Navigation


IMA Journal of Numerical Analysis Advance Access originally published online on March 7, 2005
IMA Journal of Numerical Analysis 2005 25(4):726-749; doi:10.1093/imanum/dri014
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
25/4/726    most recent
dri014v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Houston, P.
Right arrow Articles by Süli, E.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

IMA Journal of Numerical Analysis © Institute of Mathematics and its Applications 2005; all rights reserved.

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

Paul Houston1,**, Janice Robson2,*** and Endre Süli2,****

1 Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK, 2 Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK

** Email: Paul.Houston{at}mcs.le.ac.uk

*** Email: Janice.Robson{at}comlab.ox.ac.uk

**** Email: Endre.Suli{at}comlab.ox.ac.uk

We develop a one-parameter family of hp-version discontinuous Galerkin finite element methods, parameterised by {theta} [–1, 1], for the numerical solution of quasilinear elliptic equations in divergence form on a bounded open set {Omega} Rd, d ≥ 2. In particular, we consider the analysis of the family for the equation –{nabla} ·{µ(x, |{nabla}u|){nabla}u} = f(x) subject to mixed Dirichlet–Neumann boundary conditions on {partial} {Omega}. It is assumed that µ is a real-valued function, µ C( x [0, {infty})), and there exist positive constants mµ and Mµ such that mµ(t – s) ≤ µ(x, t)t – µ(x, s)s ≤ Mµ(t – s) for t ≥ s ≥ 0 and all x . Using a result from the theory of monotone operators for any value of {theta} [–1, 1], the corresponding method is shown to have a unique solution uDG in the finite element space. If u C1({Omega}) {cap} Hk({Omega}), k ≥ 2, then with discontinuous piecewise polynomials of degree p ≥ 1, the error between u and uDG, measured in the broken H1({Omega})-norm, is O(hs–1/pk–3/2), where 1 ≤ s ≤ min {p + 1, k}.

Key Words: hp-finite element methods; discontinuous Galerkin methods; quasilinear elliptic PDEs

Received on 11 June 2004. revised on 20 December 2004.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.