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

2025/01/21 by Huang, Xuehai, Tang, Zheqian · 4 citations
#65N12 #65N22 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2501.12137

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 L²-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.

Cited by

• Optimal error analysis of an interior penalty virtual element method for fourth-order singular perturbation problems
• Nonconforming Linear Element Method for a Generalized Tensor-Valued Stokes Equation with Application to the Triharmonic Equation
• Superconvergent and Divergence-Free Finite Element Methods for Stokes Equation
• Finite element conformal complexes in three dimensions

Related

• Robust residual-based a posteriori error estimators for mixed finite element methods for fourth order elliptic singularly perturbed problems
• Lowest-degree robust finite element scheme for a fourth-order elliptic singular perturbation problem on rectangular grids
• Mixed hp FEM for singularly perturbed fourth order boundary value problems with two small parameters
• Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem
• Hybrid high-order method for singularly perturbed fourth-order problems on curved domains