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

TL;DR

This paper introduces a goal-oriented error estimation framework for convection-diffusion-reaction equations using a variational multiscale approach with an orthogonal sub-grid scale method, offering computational efficiency and effective error control.

Contribution

The paper develops an orthogonal sub-grid scale variational multiscale method for goal-oriented error estimation, reducing computational cost compared to duality-based methods.

Findings

Both VMS and duality-based methods provide similar error estimates.

The VMS-based explicit approach is computationally less expensive.

Numerical tests confirm the effectiveness of the proposed error estimation techniques.

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.