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Elliptic and Parabolic Problems pp 279–290Cite as

Measure Data and Numerical Schemes for Elliptic Problems

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Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 63))

Abstract

In order to show existence of solutions for linear elliptic problems with measure data, a first classical method, due to Stampacchia, is to use a duality argument (and a regularity result for elliptic problems). Another classical method is to pass to the limit on approximate solutions obtained with regular data (converging towards the measure data). A third method is presented. It consists to pass to the limit on approximate solutions obtained with numerical schemes such that Finite Element schemes or Finite Volume schemes. This method also works for convection-diffusion problems which lead to non coercive elliptic problems with measure data. Thanks to a uniqueness result, the convergence of the approximate solutions as the mesh size vanishes is also achieved.

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Author information

Authors and Affiliations

  1. LATP, CMI, F-13453, Marseille cedex 13, France

    Thierry Gallouët

Authors
  1. Thierry Gallouët
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Editor information

Editors and Affiliations

  1. Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Catherine Bandle

  2. Ecole des hautes études en sciences sociales (EHESS), CAMS, 54, Boulevard Raspail, 75006, Paris, France

    Henri Berestycki

  3. Faculté des Sciences et Techniques, Université de Haute-Alsace, 4 rue des frères Lumières, 68093, Mulhouse Cedex, France

    Bernard Brighi

  4. Laboratoire de Gestion des Risques et Environnement, Université de Haute-Alsace, 25, rue de Chemnitz, 68200, Mulhouse, France

    Alain Brillard

  5. Angewandte Mathematik, Universität Zürich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  6. Département de mathématiques, Université de Paris-Sud, Bâtiment 425, 91405, Orsay, France

    Jean-Michel Coron

  7. Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia, 80126, Napoli, Italy

    Carlo Sbordone

  8. Department of Mathematics, Technion — Israel Institute of Technology, 32000, Haifa, Israel

    Itai Shafrir

  9. CNR-IAC, Viale del Policlinico, 137, 00161, Roma, Italy

    Vanda Valente

  10. Dipartimento di metodi e modelli matematici per le Scienze Aplicate, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161, Roma, Italy

    Giorgio Vergara Caffarelli

Additional information

Dedicated to H. Brezis in the occasion of his 60th birthday

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Gallouët, T. (2005). Measure Data and Numerical Schemes for Elliptic Problems. In: Bandle, C., et al. Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 63. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7384-9_28

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