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 $V_{h,\infty}^k$, 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.