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We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov-Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right hand side load vectors for the two methods. This new approach allows for improving well known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov-Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.
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This content will become publicly available on December 9, 2025
Comparison of variational discretizations for a convection-diffusion problem
For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare three methods: the standard linear discretization, the saddle point least square and the upwinding Petrov-Galerkin methods. We conclude that the bubble upwinding Petrov-Galerkin method is the most performant discretization for the one dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multidimensional cases.
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- Award ID(s):
- 2011615
- PAR ID:
- 10560479
- Editor(s):
- Beznea, Lucian; Putinar, Mihai
- Publisher / Repository:
- Romanian Academy, Publishing House of the Romanian Academy
- Date Published:
- Journal Name:
- Revue Roumaine Mathématiques Pures et Appliquées
- Volume:
- Tome LXIX
- Issue:
- 3-4
- ISSN:
- 0035-3965
- Page Range / eLocation ID:
- 327-351
- Subject(s) / Keyword(s):
- Petrov Galerkin upwinding convection dominated problem singularly perturbed problems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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