Mixed $hp$ FEM for singularly perturbed fourth order boundary value problems with two small parameters

TL;DR

This paper develops an $hp$ finite element method on a spectral boundary layer mesh for singularly perturbed fourth order problems with two small parameters, achieving uniform exponential convergence in energy norms.

Contribution

It introduces a mixed $hp$ FEM approach on spectral boundary layer meshes for problems with two small parameters, proving uniform exponential convergence.

Findings

Method converges exponentially in energy norm for analytic data.

Numerical examples confirm theoretical convergence rates.

Approach handles two small parameters simultaneously.

Abstract

We consider fourth order singularly perturbed boundary value problems with two small parameters, and the approximation of their solution by the $hp$ version of the Finite Element Method on the {\emph{Spectral Boundary Layer}} mesh from \cite{MXO}. We use a mixed formulation requiring only $C^{0}$ basis functions in two-dimensional smooth domains. Under the assumption of analytic data, we show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through numerical examples, including results using a stronger (balanced) norm.