A Morley-Wang-Xu element method for a fourth order elliptic singular perturbation problem

TL;DR

This paper introduces a Morley-Wang-Xu (MWX) element method for fourth order elliptic singular perturbation problems, providing sharp error analysis, optimal convergence, and solver efficiency through decoupling techniques.

Contribution

The paper proposes a novel MWX element method with a modified right hand side, enabling efficient decoupling and solver design for complex fourth order problems.

Findings

Sharp error estimates are established for the MWX method.

The method achieves optimal convergence rates in boundary layer cases.

Decoupling into simpler problems leads to efficient solvers.

Abstract

A Morley-Wang-Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods. The sharp error analysis is given for this MWX element method. And the Nitsche's technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers. An important feature of the MWX element method is solver-friendly. Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming $P_1$-$P_0$ element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method. Some numerical examples are provided to verify the theoretical results.