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Numerische Mathematik

, Volume 139, Issue 2, pp 315–348 | Cite as

Finite element methods for Darcy’s problem coupled with the heat equation

  • Christine Bernardi
  • Séréna DibEmail author
  • Vivette Girault
  • Frédéric Hecht
  • François Murat
  • Toni Sayah
Article
  • 407 Downloads

Abstract

In this article, we study theoretically and numerically the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We propose and analyze two numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization.

Mathematics Subject Classification

35K05 25B45 74S05 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Christine Bernardi
    • 1
  • Séréna Dib
    • 1
    • 2
    Email author
  • Vivette Girault
    • 1
  • Frédéric Hecht
    • 1
  • François Murat
    • 1
  • Toni Sayah
    • 2
  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie, Paris VIParis Cedex 05France
  2. 2.Laboratoire de Mathématiques et Applications (LMA), Unité de recherche “Mathématiques et Modélisation” (MM), Faculté des SciencesUniversité Saint-JosephBeirutLebanon

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