CGDG finite-element error estimates for the Mindlin plate

Let $\Omega$ be the spatial domain, $h$ the mesh size, and $k$ the polynomial order in the CGDG discretization. Let $e_w$, $\bm{e}_\theta$, $\bm{E}_\kappa$, and $\bm{e}_\gamma$ denote the exact error variables for the Mindlin plate problem, and let $e_w^h$, $\bm{e}_\theta^h$, $\bm{E}_\kappa^h$, and $\bm{e}_\gamma^h$ denote their discrete approximations. For norms over the time interval, write $L^{\infty}(X)$ for the corresponding essential-supremum-in-time norm. Assuming a smooth solution to problem~, the following error estimates hold:

$$\begin{aligned} \lVert e_w-e_w^h\rVert_{L^{\infty}(H^1(\Omega))}&\lesssim h^k,\\ \lVert\bm{e}_\theta-\bm{e}_\theta^h\rVert_{L^{\infty}(H^{\operatorname*{Grad}}(\Omega,\mathbb{R}^2))}&\lesssim h^k, \end{aligned} \qquad \begin{aligned} \lVert\bm{E}_\kappa-\bm{E}_\kappa^h\rVert_{L^{\infty}(L^2(\Omega,\mathbb{S}))}&\lesssim h^k,\\ \lVert\bm{e}_\gamma-\bm{e}_\gamma^h\rVert_{L^{\infty}(L^2(\Omega,\mathbb{R}^2))}&\lesssim h^k. \end{aligned}$$

Here $a\lesssim b$ means $a\leq Cb$, with $C$ depending only on the true solution and the final time. CGDG finite-element error-estimate conjecture. For the CGDG discretization consisting of continuous Galerkin spaces for the displacement and rotation variables and discontinuous Galerkin spaces for the stress and shear variables, all four errors converge with order $k$ in the stated norms for smooth solutions. This conjecture concerns the expected spatial convergence of the reduced-integration finite-element approximation; the supplied text gives no proof or resolution status.