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IMA Journal of Numerical Analysis Advance Access published online on June 10, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn015
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Discontinuous Galerkin methods for the biharmonic problem

Emmanuil H. Georgoulis{dagger}

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

Paul Houston{ddagger}

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

{dagger} Email: emmanuil.georgoulis{at}mcs.le.ac.uk

{ddagger} Email: paul.houston{at}nottingham.ac.uk

Received on 7 June 2007. Revised on 28 January 2008.


  Abstract

This work is concerned with the design and analysis of hp-version discontinuous Galerkin (DG) finite element methods for boundary-value problems involving the biharmonic operator. The first part extends the unified approach of Arnold et al. (2001/2002, SIAM J. Numer. Anal., 39, 1749-–1779) developed for the Poisson problem, to the design of DG methods via an appropriate choice of numerical flux functions for fourth-order problems; as an example, we retrieve the interior penalty DG method developed by Süli & Mozolevski (2007, Comput. Methods Appl. Mech. Eng., 196, 1851-–1863). The second part of this work is concerned with a new a priori error analysis of the hp-version interior penalty DG method, when the error is measured in terms of both the energy norm and the L2-norm, as well as certain linear functionals of the solution, for elemental polynomial degrees p ≥ 2. Also, provided that the solution is piecewise analytic in an open neighbourhood of each element, exponential convergence is also proved for the p-version of the DG method. The sharpness of the theoretical developments is illustrated by numerical experiments.

Key Words: discontinuous Galerkin; finite element methods; biharmonic problem; fourth order PDEs


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