Analysis of SDFEM on Shishkin triangular meshes and hybrid meshes for problems with characteristic layers

TL;DR

This paper analyzes the effectiveness of the SDFEM on Shishkin and hybrid meshes for singularly perturbed problems with layers, revealing optimal mesh strategies and confirming results through numerical experiments.

Contribution

It provides a novel analysis of SDFEM on hybrid meshes, including supercloseness properties and recommendations for mesh element types in different layer regions.

Findings

Supercloseness property of $u^I-u^N$ established.

Bilinear elements recommended for exponential layers.

Numerical experiments confirm theoretical analysis.

Abstract

In this paper, we analyze the streamline diffusion finite element method (SDFEM) for a model singularly perturbed convection-diffusion equation on a Shishkin triangular mesh and hybrid meshes. Supercloseness property of $u^I-u^N$ is obtained, where $u^I$ is the interpolant of the solution $u$ and $u^N$ is the SDFEM's solution. The analysis depends on novel integral inequalities for the diffusion and convection parts in the bilinear form. Furthermore, analysis on hybrid meshes shows that bilinear elements should be recommended for the exponential layer, not for the characteristic layer. Finally, numerical experiments support these theoretical results.