A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept
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arXiv
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| Format: | Preprint |
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2025
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| author | Codina, Ramon Gravenkamp, Hauke Khan, Sheraz Ahmed |
| author_facet | Codina, Ramon Gravenkamp, Hauke Khan, Sheraz Ahmed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept Codina, Ramon Gravenkamp, Hauke Khan, Sheraz Ahmed Numerical Analysis 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. |
| title | A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.02531 |