On convergence of upwinding Petrov-Galerkin methods for convection-diffusion
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912571689271296 |
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| author | Bacuta, Constantin |
| author_facet | Bacuta, Constantin |
| contents | We consider special upwinding Petrov-Galerkin discretizations for convection-diffusion problems. For the one dimensional case with a standard continuous linear element as the trial space and a special exponential bubble test space, we prove that the Green function associated to the continuous solution can generate the test space. In this case, we find a formula for the exact inverse of the discretization matrix, that is used for establishing new error estimates for other bubble upwinding Petrov-Galerkin discretizations. We introduce a quadratic bubble upwinding method with a special scaling parameter that provides optimal approximation order for the solution in the discrete infinity norm. % while avoiding exponential test functions. Provided the linear interpolant has standard approximation properties, we prove optimal approximation estimates in $L^2$ and $H^1$ norms. The quadratic bubble method is extended to a two dimensional convection diffusion problem. The proposed discretization produces optimal $L^2$ and $H^1$ convergence orders on subdomains that avoid the boundary layers. The tensor idea of using an efficient upwinding Petrov-Galerkin discretization along each stream line direction in combination with a standard discretizations for the orthogonal direction(s) can lead to new and efficient discretization methods for multidimensional convection dominated models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On convergence of upwinding Petrov-Galerkin methods for convection-diffusion Bacuta, Constantin Numerical Analysis 65F We consider special upwinding Petrov-Galerkin discretizations for convection-diffusion problems. For the one dimensional case with a standard continuous linear element as the trial space and a special exponential bubble test space, we prove that the Green function associated to the continuous solution can generate the test space. In this case, we find a formula for the exact inverse of the discretization matrix, that is used for establishing new error estimates for other bubble upwinding Petrov-Galerkin discretizations. We introduce a quadratic bubble upwinding method with a special scaling parameter that provides optimal approximation order for the solution in the discrete infinity norm. % while avoiding exponential test functions. Provided the linear interpolant has standard approximation properties, we prove optimal approximation estimates in $L^2$ and $H^1$ norms. The quadratic bubble method is extended to a two dimensional convection diffusion problem. The proposed discretization produces optimal $L^2$ and $H^1$ convergence orders on subdomains that avoid the boundary layers. The tensor idea of using an efficient upwinding Petrov-Galerkin discretization along each stream line direction in combination with a standard discretizations for the orthogonal direction(s) can lead to new and efficient discretization methods for multidimensional convection dominated models. |
| title | On convergence of upwinding Petrov-Galerkin methods for convection-diffusion |
| topic | Numerical Analysis 65F |
| url | https://arxiv.org/abs/2509.04703 |