Analysis of Galerkin and streamline-diffusion FEMs on piecewise equidistant meshes for turning point problems exhibiting an interior layer

TL;DR

This paper analyzes higher order finite element methods, including streamline-diffusion FEM, on specialized meshes for singularly perturbed problems with interior layers, providing uniform error estimates and confirming results through numerical experiments.

Contribution

It offers new uniform error estimates for Galerkin and streamline-diffusion FEMs on layer-adapted meshes for interior layer problems, with theoretical proofs and numerical validation.

Findings

Error estimates are uniform with respect to the perturbation parameter.

Numerical experiments confirm the theoretical error bounds.

Streamline-diffusion FEM effectively captures interior layers.

Abstract

We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted piecewise equidistant meshes proposed by Sun and Stynes. We also study the streamline-diffusion finite element method (SDFEM) for such problems. For these methods error estimates uniform with respect to $\varepsilon$ are proven in the energy norm and in the stronger SDFEM-norm, respectively. Numerical experiments confirm the theoretical findings.