Uniform discrete Korn control for the lowest-order HHO method

Determine whether, for polynomial degree k=0 and homogeneous Dirichlet face unknowns, the bilinear form a(v_h) controls uniformly in the mesh size h the corresponding quantity obtained by replacing the cellwise symmetric-gradient seminorm with the cellwise broken-gradient seminorm, namely whether there exists an h-independent constant such that the form defined by the cellwise symmetric gradients and projected face jumps controls the analogous form containing \|\nabla v_T\|_T^2.

Background

The paper’s analysis assumes polynomial degree k\geq1 because the face-space stabilization controls projected face jumps and because the relevant local Korn estimate fails for k=0. For k=0, face unknowns are constant on each face, so stabilization controls only the mean jump. The authors therefore introduce a bilinear form combining the cellwise symmetric-gradient seminorm with the h-scaled projected face-jump seminorm and ask whether it provides uniform control of the broken cellwise gradient.

The authors report that the answer depends on the mesh family: the form is singular on the considered criss-cross simplicial meshes, its Korn constant decays like O(h^2) on quadrilateral meshes, and it appears uniform on the tested polygonal meshes. However, these observations are presented through numerical eigenvalue measurements rather than a general h-uniform theorem, leaving the general discrete Korn characterization unresolved.