The objective of this paper is to show that the approximate solution, by the finite volumes method, converges to the renormalized solution of elliptic problems with measure data. The methods used are a priori estimates and density arguments. In the first part, we recall formulas and give some notations which are useful for the next of the work. It is also mentioned some definitions and properties on Partial Differentials Equations. In the second part we show the bases principle of the main methods of discretization, more precisely, the finite volume method. In the third part, we study a no coercive elliptic convection-diffusion equation with measure data. In our case, we take a diffuse measure data instead of L1-data. The main originality in the present work is that we pass to the limit in a “renormalized discrete version”. A first difficulty is to establish a discrete version of the estimate on the energy. The second difficulty is to deal with the diffuse measure data. By adapting the strategy developed in the finite volume method, we state and show our main result: the approximate solution converges to the unique renormalized solution. This work ends with a conclusion.
| Published in | Pure and Applied Mathematics Journal (Volume 12, Issue 1) |
| DOI | 10.11648/j.pamj.20231201.11 |
| Page(s) | 1-11 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
lliptic Problem, Measure Data, Renormalized Solutions, Finite Volume Scheme
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APA Style
Arouna Ouédraogo, Wendlassida Basile Yaméogo. (2023). Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data. Pure and Applied Mathematics Journal, 12(1), 1-11. https://doi.org/10.11648/j.pamj.20231201.11
ACS Style
Arouna Ouédraogo; Wendlassida Basile Yaméogo. Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data. Pure Appl. Math. J. 2023, 12(1), 1-11. doi: 10.11648/j.pamj.20231201.11
AMA Style
Arouna Ouédraogo, Wendlassida Basile Yaméogo. Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data. Pure Appl Math J. 2023;12(1):1-11. doi: 10.11648/j.pamj.20231201.11
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Download@article{10.11648/j.pamj.20231201.11,
author = {Arouna Ouédraogo and Wendlassida Basile Yaméogo},
title = {Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data},
journal = {Pure and Applied Mathematics Journal},
volume = {12},
number = {1},
pages = {1-11},
doi = {10.11648/j.pamj.20231201.11},
url = {https://doi.org/10.11648/j.pamj.20231201.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20231201.11},
abstract = {The objective of this paper is to show that the approximate solution, by the finite volumes method, converges to the renormalized solution of elliptic problems with measure data. The methods used are a priori estimates and density arguments. In the first part, we recall formulas and give some notations which are useful for the next of the work. It is also mentioned some definitions and properties on Partial Differentials Equations. In the second part we show the bases principle of the main methods of discretization, more precisely, the finite volume method. In the third part, we study a no coercive elliptic convection-diffusion equation with measure data. In our case, we take a diffuse measure data instead of L1-data. The main originality in the present work is that we pass to the limit in a “renormalized discrete version”. A first difficulty is to establish a discrete version of the estimate on the energy. The second difficulty is to deal with the diffuse measure data. By adapting the strategy developed in the finite volume method, we state and show our main result: the approximate solution converges to the unique renormalized solution. This work ends with a conclusion.},
year = {2023}
}
TY - JOUR T1 - Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data AU - Arouna Ouédraogo AU - Wendlassida Basile Yaméogo Y1 - 2023/03/02 PY - 2023 N1 - https://doi.org/10.11648/j.pamj.20231201.11 DO - 10.11648/j.pamj.20231201.11 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 1 EP - 11 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20231201.11 AB - The objective of this paper is to show that the approximate solution, by the finite volumes method, converges to the renormalized solution of elliptic problems with measure data. The methods used are a priori estimates and density arguments. In the first part, we recall formulas and give some notations which are useful for the next of the work. It is also mentioned some definitions and properties on Partial Differentials Equations. In the second part we show the bases principle of the main methods of discretization, more precisely, the finite volume method. In the third part, we study a no coercive elliptic convection-diffusion equation with measure data. In our case, we take a diffuse measure data instead of L1-data. The main originality in the present work is that we pass to the limit in a “renormalized discrete version”. A first difficulty is to establish a discrete version of the estimate on the energy. The second difficulty is to deal with the diffuse measure data. By adapting the strategy developed in the finite volume method, we state and show our main result: the approximate solution converges to the unique renormalized solution. This work ends with a conclusion. VL - 12 IS - 1 ER -
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