High-resolution solutions of nonlinear, advection-dominated problems using a generalized finite element method

TL;DR

This paper introduces a generalized finite element method with solution-specific enrichments to effectively stabilize and accurately solve advection-dominated problems like the Burgers equation, reducing oscillations and improving accuracy.

Contribution

It presents a novel enrichment approach within the finite element framework tailored for advection-dominated problems, demonstrating improved stability and accuracy over traditional methods.

Findings

Oscillations are naturally alleviated with the proposed method.

Significant error reduction compared to standard finite element methods.

Effective capture of boundary layers and shocks on coarse grids.

Abstract

Traditional finite element approaches are well-known to introduce spurious oscillations when applied to advection-dominated problems. We explore alleviation of this issue from the perspective of a generalized finite element formulation, which enables stabilization through an enrichment process. The presented work uses solution-tailored enrichments for the numerical solution of the one-dimensional, unsteady Burgers equation. Mainly, generalizable exponential and hyperbolic tangent enrichments effectively capture local, steep boundary layer/shock features. Results show natural alleviation of oscillations and return smooth numerical solutions over coarse grids. Additionally, significantly improved error levels are observed compared to Lagrangian finite element methods.