On the Penalty term for the Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation
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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 Δ^2 u = f into two second order problems by introducing an auxiliary variable v = -Δ u. We prove that choosing the penalty term α_k = σ_0 |e_k|^-1p^2 < σ_0 |e_k|^-3 p^2 for a sufficiently large σ_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.
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