Notes on Finite Element Discretization for a Model Convection-Diffusion Problem

TL;DR

This paper compares finite element discretizations for a convection-diffusion problem in the singularly perturbed regime, analyzing oscillations and error norms to improve numerical stability and accuracy.

Contribution

It introduces a saddle point least squares discretization with quadratic test functions and relates it to existing methods like up-winding Petrov-Galerkin and stream-line diffusion.

Findings

Saddle point least squares reduces non-physical oscillations.

Comparison shows differences in error norms between methods.

Results suggest extensions to multidimensional problems.

Abstract

We present recent finite element numerical results on a model convection-diffusion problem in the singular perturbed case when the convection term dominates the problem. We compare the standard Galerkin discretization using the linear element with a saddle point least square discretization that uses quadratic test functions, trying to control and explain the non-physical oscillations of the discrete solutions. We also relate the up-winding Petrov-Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. Some results can be extended to the multidimensional case in order to come up with efficient approximations for more general singular perturbed problems, including convection dominated models.