Discontinuous Galerkin Methods for the Biharmonic Problem

Georgoulis, Emmanuil H.; Houston, Paul

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, Brezzi, Cockburn & Marini (SIAM J. Numer. Anal. 39, 5 (2001/02), 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 Suli & Mozolevski (Comput. Methods Appl. Mech. Engrg. 196, 13-16 (2007), 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 L2-norm, as well certain linear functionals of the solution, for elemental polynomial degrees $p\ge 2$. Also, provided that the solution is piecewise analytic in an open neighbourhood of each element, exponential convergence is also proven for the p-version of the DG method. The sharpness of the theoretical developments is illustrated by numerical experiments.

Citation

Georgoulis, E. H., & Houston, P. Discontinuous Galerkin Methods for the Biharmonic Problem. Manuscript submitted for publication

• Journal Article Type: Article
• Deposit Date: Oct 23, 2007
• Peer Reviewed: Not Peer Reviewed
• Keywords: Discontinuous, Galerkin Methods, Biharmonic, finite element methods
• Public URL: https://nottingham-repository.worktribe.com/output/1026187
• Additional Information: Article has been accepted for publication in IMA Journal of Numerical Analysis ©: 2007 Institute of Mathematics and its Applications. Published by Oxford University Press [on behalf of IMA]. All rights reserved.