A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems

TL;DR

This paper tackles goal-oriented a posteriori error estimation for the stationary convection-diffusion-reaction equation using a stabilized finite element method within a variational multiscale framework. It develops two estimators under the orthogonal subgrid scales (OSGS) stabilization: an explicit post-processing estimator and an implicit adjoint-based estimator, showing they yield similar global error trends, with the explicit method being computationally cheaper. The authors prove (theoretically) the equivalence of the estimators under exact SGS and demonstrate, through one- and two-dimensional numerical tests including strong boundary layers and an L-shaped domain, that both estimators achieve near-unit effectivity and accurate reproduction of the true quantity of interest. The work highlights the limitations of global error norms for convection-dominated problems and provides SGS-based, goal-oriented indicators that are suitable for adaptive refinement, offering a practical and efficient path for reliable simulations of CDR systems.

Abstract

This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for error estimation is based on the variational multiscale (VMS) concept, where the solution is decomposed into resolved (finite element) and unresolved (sub-grid) scales. In this work, we propose an orthogonal sub-grid scale (OSGS) method for a goal-oriented error estimation in VMS discretizations. In the OSGS approach, the space of the sub-grid scales (SGSs) is orthogonal to the finite element space. The error is estimated in the quantity of interest, given by the linear functional $Q(u)$ of the unknown $u$. If the SGS $u'$ is estimated, the error in the quantity of interest can be approximated by $Q(u')$. Our approach is compared with a duality-based a posteriori error estimation method, which requires the solution of an additional auxiliary problem. The results indicate that both methods yield similar error estimates, whereas the VMS-based explicit approach is computationally less expensive than the duality-based implicit approach. Numerical tests demonstrated the effectiveness of our proposed error estimation techniques in terms of the quantity of interest functionals.

Figures (10)

• Figure 1: Results of example \ref{['example-1']}: Error convergence in the quantity of interest functional $Q$ compared to the goal-oriented error estimator (a); global effectivity index (b).
• Figure 2: Results of example \ref{['example-2']}: Error convergence in the quantity of interest $Q$ compared to the goal-oriented error estimator (a); global effectivity index (b).
• Figure 3: Results of example \ref{['example-2']}: (a): Numerical solution of the primal problem \ref{['strprim']}. (b): Numerical solution of the dual problem \ref{['strdual']}.
• Figure 4: Results of example \ref{['example-2']}: Local contribution of the explicit error estimator $\eta_1$ (a) and implicit error estimator $\eta_2$ (b).
• Figure 5: Results of example \ref{['example-3']}: Error convergence in the quantity of interest $Q$ compared to the goal-oriented error estimator for the strong boundary layer problem (a); global effectivity index (b).