Conjectured convergence for weakly symmetric mixed finite elements for Mindlin plates

Consider the weakly symmetric Mindlin plate formulation with unknowns $e_w$, $\bm{e}_{\theta}$, $\bm{E}_{\kappa}$, $\bm{e}_{\gamma}$, and $\bm{E}_r$, and discrete approximations $e_w^h$, $\bm{e}_{\theta}^h$, $\bm{E}_{\kappa}^h$, $\bm{e}_{\gamma}^h$, and $\bm{E}_r^h$. The notation $a\lesssim b$ denotes $a\leq Cb$.

Weak-symmetry convergence conjecture. Assuming a smooth solution to the Mindlin plate problem, the error estimates

$$\lVert e_w-e_w^h\rVert_{L^{\infty}(L^2)}\lesssim h^k,$$ $$\lVert\bm{e}_{\theta}-\bm{e}_{\theta}^h\rVert_{L^{\infty}(L^2)}\lesssim h^k,$$ $$\lVert\bm{E}_r-\bm{E}_r^h\rVert_{L^{\infty}(L^2)}\lesssim h^k,$$ $$\lVert\bm{E}_{\kappa}-\bm{E}_{\kappa}^h\rVert_{L^{\infty}(L^2)}\lesssim h^k,$$ $$\lVert\bm{e}_{\gamma}-\bm{e}_{\gamma}^h\rVert_{L^{\infty}(L^2)}\lesssim h^k$$

should hold.

The conjecture combines known convergence analyses for weakly symmetric elastodynamics and mixed finite elements for the wave equation. The stated dynamic estimates remain unproved in the source.