Optimal Adaptivity for the SUPG Finite Element Method

TL;DR

This paper develops an adaptive mesh refinement algorithm for the SUPG finite element method, ensuring convergence at optimal rates for convection-dominated problems by leveraging robust a posteriori error estimators.

Contribution

It introduces an adaptive algorithm for SUPG that guarantees asymptotically optimal convergence rates, a novel approach in finite element analysis for convection-dominated issues.

Findings

Convergence of SUPG solutions at optimal rates.

Effective use of a posteriori error estimators.

Validated adaptive mesh refinement improves solution accuracy.

Abstract

For convection dominated problems, the streamline upwind Petrov--Galerkin method (SUPG), also named streamline diffusion finite element method (SDFEM), ensures a stable finite element solution. Based on robust a posteriori error estimators, we propose an adaptive mesh-refining algorithm for SUPG and prove that the generated SUPG solutions converge at asymptotically optimal rates towards the exact solution.