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arXiv · 2505.02946

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

Sheraz Ahmed Khan·Ramon Codina·Hauke Gravenkamp
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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.

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Sheraz Ahmed Khan, Ramon Codina, Hauke Gravenkamp. 2025-05-05. A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems. https://arxiv.org/abs/2505.02946

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