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IMA Journal of Numerical Analysis Advance Access published online on March 7, 2005
IMA Journal of Numerical Analysis, doi:10.1093/imanum/dri014
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IMA Journal of Numerical Analysis © Institute of Mathematics and its Applications 2005; all rights reserved.
Received June 11, 2004
Revised December 20, 2004

Article

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

Paul Houston 1*, Janice Robson 2, and Endre Süli 2

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

* To whom correspondence should be addressed.
Paul Houston, E-mail: Paul.Houston{at}mcs.le.ac.uk


  Abstract

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 ({{Omega}macr} 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 {{Omega}macr}. 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}.

Keywords: hp-finite element methods; discontinuous Galerkin methods; quasilinear elliptic PDEs.
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