On the Penalty term for the Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

K Balaje, P Danumjaya

Published 2019-11-28Version 1

In this paper, we present an analysis of the effect of penalty term in the mixed discontinuous Galerkin finite element method for the biharmonic equation. We split the biharmonic problem $\Delta^2 u = f$ into two second order problems by introducing an auxiliary variable $v = -\Delta u$. We prove that choosing the penalty term $\alpha_k = \sigma_0 |e_k|^{-1}p^2 < \sigma_0 |e_k|^{-3} p^2$ for a sufficiently large $\sigma_0$, ensures optimal rate of convergence in the $L^2$ and the energy norm for the approximations $u_h$ and $v_h$. Finally, we present numerical experiments to validate our theoretical results.