| Peer-Reviewed

Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data

Received: 14 January 2023     Accepted: 20 February 2023     Published: 2 March 2023
Views:       Downloads:
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.

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.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

lliptic Problem, Measure Data, Renormalized Solutions, Finite Volume Scheme

References
[1] M. Ben Cheick and O. Guibé, Nonlinear and non- coercive elliptic problems with integrable data. Adv. Math. Sci. Appl. 16 n◦1, 275 − 297, 2006.
[2] L. Boccardo and T. Gallouët, On some nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal., 87, 149-169, 1989.
[3] M. Botti, D. A. Di Pietro, O. Le Maˆıtre, P. Sochala. Numerical Approximation of Poroelasticity with Random Coefficients Using Polynomial Chaos and High-Order Polyhedral Methods. Comput. Meth. Appl. Mech. Eng. 361 (2020). https://dx.doi.org/10.1016/j.cma.2019.112736.
[4] M. Botti, D. A. Di Pietro, P. Sochala, A Hybrid High-Order discretization method for nonlinear poroelasticity. Comput. Meth. Appl. Math. (2019). https://doi.org/10.1515/cmam-2018-0142.
[5] J. Carrillo and M. Chipot, On some nonlinear elliptic equations involving derivatives of the nonlinearity. Proc. Roy. Soc. Edinburgh Sect A100, n◦3−4, 281-294, 1985.
[6] D. Castanon Quiroz, D. A. Di Pietro. A Hybrid High- Order Method for the Incompressible Navier-Stokes Problem Robust for Large Irrotational Body Forces (2020). https://dx.doi.org/10.1016/j.camwa.2019.12.005.
[7] C. Chainais-Hillairet and J. Droniou, Finite volume schemes for noncoercive elliptic problems with Neumann boundary conditions. IMA J. Numer. Anal. 31, n◦1, 61- 85, 2011.
[8] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28(4), 741-808, 1999.
[9] D. A. Di Pietro, J. Droniou, The Hybrid High-Order method for polytopal meshes, Modeling, Simulation and Application, vol. 19, Springer International Publishing, 2020.
[10] J. Droniou, T. Gallouët and R. Herbin, A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal., 6, 1997-2031, 2003.
[11] R. Eymard, T. Gallouët and R. Herbin, Convergence of finite volume approximations to the solutions of semilinear convection diffusion reaction equations. Numer. Math., 82, 91-116 (1999).
[12] O. Guibé and A. Mercaldo, Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Commun. Pure Appl. Anal. 7, n◦1, 163- 192, 2008.
[13] A. Larcher and J. C. Latché, Convergence analysis of a finite element-finite volume scheme for a RANS turbulencemodel. InstitutdeRadioprotectionetdeSureté Nucléaire (IRSN), France, 2012.
[14] S. Leclavier, Finite volume scheme and renormalized solutions for a noncoercive elliptic problem with L1- data, Comput. Methods Appl. Math. 17, No. 1, 85-104 (2017).
[15] F. Murat, Soluciones renormalizadas de EDP elipticas no lineales. Technical Report R93023, Laboratoire d’Analyse Numérique, Paris VI, 1993.
[16] A. Ouédraogo, T. Valea and W. B. Yaméogo, Renormalized solutions for a non-coercive elliptic problem with measure data. Submitted.
Cite This Article
  • 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

    Copy | Download

    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

    Copy | Download

    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

    Copy | 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}
    }
    

    Copy | Download

  • 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  - 

    Copy | Download

Author Information
  • Department of Mathematics, Faculty of Sciences and Technology, University Norbert Zongo, Koudougou, Burkina Faso

  • Department of Mathematics, Faculty of Sciences and Technology, University Norbert Zongo, Koudougou, Burkina Faso

  • Sections