Symmetric-Tensor Distributional Mixed Method for Fourth-Order Elliptic Singular Perturbation Problem

A symmetric-tensor distributional mixed method for a fourth-order elliptic singular perturbation problem is developed in this paper. The moment variable is approximated by normal-normal continuous symmetric tensor elements, while the scalar variable is represented by an $H^1$-nonconforming virtual element space coupled with a polynomial multiplier on interior subsimplices of codimension two. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form of the method is also equivalent to stabilization-free weak Galerkin and $H^2$-nonconforming virtual element methods. In two dimensions, a close connection of the distributional mixed method to the classical Hellan-Herrmann-Johnson (HHJ) method is established, by naturally identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Thus the proposed method extends the two-dimensional HHJ method to arbitrary spatial dimensions. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates.