Analysis of the SDFEM in a modified streamline diffusion norm for singularly perturbed convection diffusion problems

TL;DR

This paper analyzes the stabilized finite element method for singularly perturbed convection diffusion problems, focusing on how to choose stabilization parameters to achieve uniform estimates and stability.

Contribution

It introduces a modified streamline diffusion norm and discusses optimal stabilization parameters for improved uniform estimates in SDFEM solutions.

Findings

Proper stabilization parameters ensure stability and accuracy.

Uniform estimates are achieved in a stronger norm than the ε-energy norm.

Numerical results confirm the effectiveness of the proposed parameters.

Abstract

In this paper we consider a model singularly perturbed convection diffusion problem which is solved by a streamline diffusion finite element method (SDFEM) on a Shishkin rectangular mesh. To put insight into the influences of stabilization parameters on SDFEM's solutions, we discuss how to obtain the uniform estimates in the streamline diffusion norm which have not been analyzed up to now. By decreasing the standard stabilization parameters properly near the exponential layers, we obtain the uniform estimates in a norm, which is stronger than the $\varepsilon-$energy norm and weaker than the standard streamline diffusion norm. Numerical experiments show that the presented stabilization parameters provide enough stability for the numerical schemes and yield same accurate solutions as the standard ones.