{
  "id": "2505.02531",
  "version": "v1",
  "published": "2025-05-05T10:12:34.000Z",
  "updated": "2025-05-05T10:12:34.000Z",
  "title": "A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept",
  "authors": [
    "Ramon Codina",
    "Hauke Gravenkamp",
    "Sheraz Ahmed Khan"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-05T10:12:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "posteriori error estimate",
      "variational multiscale concept",
      "finite element approximation",
      "convection-diffusion-reaction equation",
      "sub-grid scale"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}