Dual weighted residual based error control for nonstationary convection-dominated equations: potential or ballast?
For researchers in numerical methods for convection-dominated PDEs, this work provides a critical assessment of the DWR approach, showing it offers limited potential for error control in this context.
The paper investigates whether the Dual Weighted Residual (DWR) method can improve the accuracy of stabilized finite element approximations for nonstationary convection-dominated problems, finding that strict application of DWR does not yield significant benefits and may be counterproductive.
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.