Dual weighted residual based error control for nonstationary convection-dominated equations: potential or ballast?

TL;DR

This paper investigates the effectiveness of the Dual Weighted Residual (DWR) approach in improving error control for nonstationary convection-dominated equations, emphasizing its potential rather than computational efficiency.

Contribution

It explores the application of DWR to stabilized finite element methods for convection-dominated problems, highlighting its potential for error estimation over computational savings.

Findings

DWR enhances the quality of numerical approximations.

Strict DWR application impacts error control more than cost reduction.

Potential of DWR is emphasized over computational efficiency.

Abstract

Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.