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

Abstract

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 codimension-two subsimplices. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form is further shown to be equivalent to stabilization-free weak Galerkin and H^2-nonconforming virtual element formulations. In two dimensions, we establish a close connection between the distributional mixed method and the classical Hellan-Herrmann-Johnson (HHJ) method by identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Consequently, the proposed method extends the two-dimensional HHJ framework to any spatial dimension d >= 2. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates. A two-dimensional adaptive constant-load benchmark on an L-shaped polygonal domain tests the method on a non-manufactured nonsmooth problem and shows mesh concentration near the reentrant corner and, for small epsilon, boundary refinement at the expected O(epsilon) scale.

26 pages, 2 figures, 3 tables