An Optimal IPDG Scheme for the Biharmonic Equation

This paper presents an optimal interior penalty discontinuous Galerkin (IPDG) scheme for the planar biharmonic equation using piecewise polynomials of degree $k=3$ or $4$. In standard IPDG methods, large penalty parameters force the discrete solution into an overconstrained space, severely degrading accuracy—phenomenon known as numerical locking. To overcome this, our method enforces only vertex continuity and projects the jumps of the function and its normal derivative onto ${\mathcal{P}}^{k-3}$ and ${\mathcal{P}}^{k-2}$, respectively. Consequently, as the penalty parameters tend to infinity, the discrete solution is forced to lie in a constrained subspace $V_{h,\mathrm{\infty }}^k$, which we identify as the optimal nonconforming finite element space $B_h^k$. This intrinsic connection fundamentally eliminates numerical locking. We prove optimal error estimates of $\mathcal{O}(h^{k-1})$ in a mesh-dependent energy norm and $\mathcal{O}(h^{k+1})$ in the $L^2$ norm. Numerical experiments on both convex and L-shaped domains confirm that the proposed scheme is robust and entirely locking-free, even for extremely large penalty parameters.