Discontinuous Galerkin methods for the biharmonic problem

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.