A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept

TL;DR

This work develops a posteriori error estimates for the stationary convection-diffusion-reaction equation within the variational multiscale framework using orthogonal sub-grid scales (OSGS). The estimator leverages a stabilized norm controlled by stabilization parameters, decomposing SGSs into interior and edge contributions and deriving bounds via two strategies (Verfürth-style and John–Novo-style) and complementary numerical validation. Numerical experiments across convection- and diffusion-dominated regimes, strong boundary layers, and an L-shaped domain show that the OSGS-based estimator achieves effectivity indices near 1 and closely tracks the stabilized-norm error, often outperforming the ASGS variant. Overall, the OSGS-based a posteriori estimator offers accurate, computationally efficient error control for stabilized FE approximations of CDRE problems, with clear practical impact for adaptive mesh refinement in convection-dominated flows.

Abstract

In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family.

Figures (9)

• Figure 1: Results of example \ref{['ex1']}. The figures on the left represent the error convergence in the $L^2$ norm, the stabilized norm and the APEE, the ones in the right represent the global effectivity index. Top: OSGS method; bottom: ASGS method.
• Figure 2: Results of example \ref{['ex1']}. The figures on the left present the SGSs error estimator and on the right the stabilized norm error contribution. Top: OSGS method; bottom: ASGS method.
• Figure 3: Results of example \ref{['example-2']}. Left: The error convergence rates of the $L^2$ norm, the stabilized norm and the APEE for the diffusion-dominated problem. Right: The global effectivity index for the diffusion-dominated problem. Top: OSGS method; bottom: ASGS method.
• Figure 4: Results of example \ref{['example-2']}. The figures on the left present the SGSs error estimator and on the right the stabilized norm error contribution. Top: OSGS method; bottom: ASGS method.
• Figure 5: Results of example \ref{['example-3']}. Left: The error convergence rate. Right: The global effectivity index. Top: OSGS method. Bottom: ASGS method.

Theorems & Definitions (28)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Theorem 1: Quasi inf-sup stability for $B_{\rm stab}$ in $V$
• proof
• Corollary 1: inf-sup stability for $B_{\rm stab}$ in $V_h$
• proof
• Theorem 2: Convergence
• Lemma 2: boundedness of $B$