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

TL;DR

This paper develops a new a posteriori error estimator for the convection-diffusion-reaction equation using the variational multiscale method, improving error control in convection-dominated problems.

Contribution

It introduces an a posteriori error estimate based on the scaled norm of sub-grid scales within the variational multiscale framework, enhancing error assessment for convection-dominated equations.

Findings

The proposed estimator reliably measures error in convection-dominated problems.

Numerical tests demonstrate the estimator's effectiveness compared to existing methods.

The method provides control over the convective term in the error estimate.

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.