Robust and Optimal Mixed Methods for a Fourth-Order Elliptic Singular Perturbation Problem

TL;DR

This paper develops robust and optimal mixed numerical methods for a complex fourth-order elliptic singular perturbation problem, providing theoretical error estimates and validating them through numerical experiments.

Contribution

It introduces a new mixed method based on a second-order system that does not rely on Nitsche's technique, with proven robustness and optimality.

Findings

Derived robust and optimal error estimates.

Established connections to other discrete methods.

Validated theoretical results with numerical experiments.

Abstract

A series of robust and optimal mixed methods based on two mixed formulations of the fourth-order elliptic singular perturbation problem are developed in this paper. First, a mixed method based on a second-order system is proposed without relying on Nitsche's technique or interpolations. Robust and optimal error estimates are derived using an $L^2$-bounded interpolation operator for tensors. Then, its connections to other discrete methods, including weak Galerkin methods and a mixed finite element method based on a first-order system, are established. Finally, numerical experiments are provided to validate the theoretical results.