Numerical analysis of a singularly perturbed 4th order problem with a shift term

TL;DR

This paper develops a numerical method for a singularly perturbed 4th order differential problem with a shift term, achieving optimal convergence and supercloseness on layer-adapted meshes, supported by theoretical analysis and numerical tests.

Contribution

It introduces a new numerical approach for a complex 4th order problem with shift term, including a formal solution decomposition and stability analysis.

Findings

Achieves supercloseness and optimal convergence on layer-adapted meshes.

Provides stability estimates and a formal solution decomposition.

Numerical examples confirm theoretical results.

Abstract

We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.