@article { , title = {Discontinuous Galerkin Methods for the Biharmonic Problem}, 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.}, issn = {0272-4979}, journal = {IMA Journal of Numerical Analysis}, publicationstatus = {Submitted}, publisher = {Oxford University Press}, url = {https://nottingham-repository.worktribe.com/output/1026187}, keyword = {Discontinuous, Galerkin Methods, Biharmonic, finite element methods}, author = {Georgoulis, Emmanuil H. and Houston, Paul} }