Numerical study of goal-oriented error control for stabilized finite element methods

TL;DR

This paper integrates the Dual Weighted Residual method with stabilized finite element techniques to improve goal-oriented error control in convection-dominated problems, enabling accurate resolution of layers and fronts.

Contribution

It introduces a higher order dual problem discretization within the DWR framework for stabilized FEM, enhancing robustness and accuracy in error estimation.

Findings

Effective resolution of layers and sharp fronts.

Reduction of spurious oscillations.

Robust error control for quantities of interest.

Abstract

The efficient and reliable approximation of convection-dominated problems continues to remain a challenging task. To overcome the difficulties associated with the discretization of convection-dominated equations, stabilization techniques and a posteriori error control mechanisms with mesh adaptivity were developed and studied in the past. Nevertheless, the derivation of robust a posteriori error estimates for standard quantities and in computable norms is still an unresolved problem and under investigation. Here we combine the Dual Weighted Residual (DWR) method for goal-oriented error control with stabilized finite element methods. By a duality argument an error representation is derived on that an adaptive strategy is built. The key ingredient of this work is the application of a higher order discretization of the dual problem in order to make a robust error control for user-chosen quantities of interest feasible. By numerical experiments in 2D and 3D we illustrate that this interpretation of the DWR methodology is capable to resolve layers and sharp fronts with high accuracy and to further reduce spurious oscillations.