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

TL;DR

This paper investigates how the choice of penalty term affects the convergence of the mixed Discontinuous Galerkin Finite Element Method for the biharmonic equation, demonstrating that an appropriate penalty improves convergence rates.

Contribution

It introduces a modified penalty term proportional to |e_k|^{-1} that achieves optimal convergence rates for both linear and quadratic elements in the method.

Findings

Optimal convergence achieved with the new penalty term

Numerical experiments validate theoretical predictions

Improved performance for piecewise linear elements

Abstract

In this paper, we present a study on the effect of penalty term in the mixed Discontinuous Galerkin Finite Element Method for the biharmonic equation proposed by \cite{gudi2008mixed}. The proposed mixed Discontinuous Galerkin Method showed sub-optimal rates of convergence for piecewise quadratic elements and no significant convergence rates for piecewise linear elements. We show that by choosing the penalty term proportional to $|e_k|^{-1}$ instead of $|e_k|^{-3}$, ensures an optimal rate of convergence for the approximation, including for piecewise linear elements. Finally, we present numerical experiments to validate our theoretical results.