Balanced norm estimates for $rp$-Finite Element Methods applied to singularly perturbed fourth order boundary value problems

TL;DR

This paper proves robust exponential convergence of $rp$-Finite Element Methods for singularly perturbed fourth order boundary value problems using a balanced norm, applicable in 1-D and 2-D with spectral boundary layer meshes.

Contribution

It introduces a new balanced norm analysis for $rp$-FEMs, demonstrating robust exponential convergence for complex boundary value problems in multiple dimensions.

Findings

Robust exponential convergence in the balanced norm.

Robust exponential convergence in the maximum norm.

Effective application of spectral boundary layer meshes.

Abstract

We establish robust exponential convergence for $rp$-Finite Element Methods (FEMs) applied to fourth order singularly perturbed boundary value problems, in a \emph{balanced norm} which is stronger than the usual energy norm associated with the problem. As a corollary, we get robust exponential convergence in the maximum norm. $r p$ FEMs are simply $p$ FEMs with possible repositioning of the (fixed number of) nodes. This is done for a $C^1$ Galerkin FEM in 1-D, and a $C^0$ mixed FEM in 2-D over domains with smooth boundary. In both cases we utilize the \emph{Spectral Boundary Layer} mesh.