Regularity shift for curved three-dimensional mesh elements

Establish a rigorous proof that the uniform $L^2$-to-$H^2$ regularity shift required for element-wise Neumann problems on curved three-dimensional mesh elements holds when all edges of the elements meet at angles strictly less than $pi$.

Background

The multiscale hybrid-mixed method assumes that each element inside the heterogeneous obstacle admits a uniform $L^2$-to-$H^2$ regularity estimate for Neumann problems. The paper explains that this estimate is immediate for convex elements and can be obtained in two dimensions under a suitable angle condition for curved polygons.

For three-dimensional curved elements, the paper suggests that an analogous result should hold under comparable geometric assumptions, but identifies the absence of a rigorous proof. Establishing this regularity result would validate the mesh assumption used in the convergence analysis for a broader class of three-dimensional curved meshes.