A mixed finite element method for the Babuška paradox using only discrete geometry
The classical Babuška paradox shows that solutions of simply supported plate problems on polygonal approximations of a curved domain may converge to an unintended limit. We develop a boundary-corrected $H({\rm divdiv};\mathbb S)$-$L^2$ mixed finite element method for the simply supported Kirchhoff-Love plate problem. The correction uses only the discrete boundary geometry. Introducing the bending moment as an independent unknown allows the condition $M_{nn}=0$ to be imposed directly, while an edgewise mean constraint on the effective shear suppresses the leading geometric inconsistency. This constraint improves the boundary consistency error from $\mathcal O(h^{1/2})$ to $\mathcal O(h^{3/2})$. The associated boundary corrections act in the kernel of $\rm{divdiv}$, leaving the discrete equilibrium equation unchanged and yielding a uniformly stable scheme. Under suitable regularity assumptions, we prove $L^2$-error estimates of order $h^{3/2}$ for the bending moment and the broken Hessian of the postprocessed displacement, and of order $h^2$ for the displacement. The analysis covers multiply connected domains and polygonal approximations whose boundaries may cross the physical boundary. Numerical experiments confirm these rates and the improvement over the uncorrected method.