A stabilized finite element formulation for advection-diffusion using the generalized finite element framework

TL;DR

This paper introduces a generalized finite element method with exponential enrichment functions to effectively solve advection-diffusion problems with steep gradients and high Peclet numbers, avoiding traditional stabilization issues.

Contribution

It develops an enriched finite element formulation that improves solution stability without needing stabilization parameters, applicable to complex geometries.

Findings

Smooth solutions for high Peclet numbers up to 25

Equivalent to stabilized Galerkin/least-squares methods

Effective for complex geometries using a global-local approach

Abstract

The following work presents a generalized (extended) finite element formulation for the advection-diffusion equation. Using enrichment functions that represent the exponential nature of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Peclet numbers (up to Pe = 25) in one and two-dimensions. As opposed to traditional stabilized methods that require the construction of stability parameters and stabilization terms, the present work avoids numerical instabilities by improving the classical Galerkin solution with an enrichment function. To contextualize this method among other stabilized methods, we show by decomposition of the solution (in a multiscale manner) an equivalence to both Galerkin/least-squares type methods and those that use bubble functions. This work also presents a strategy for constructing the enrichment function for problems with complex geometries by employing a global-local approach.