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\textbf{\Large Finite volume methods}
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\textbf{\Large for non smooth solutions of diffusion models;}
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\textbf{\Large Application to imperfect contact problems}
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\textbf{\large Philippe ANGOT}

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{\sl Universit\'e de la M\'editerran\'ee} \\
{\sl I.R.P.H.E. Ch\^ateau-Gombert, Equipe Math\'ematiques Num\'eriques} \\
{\sl La Jet\'ee, Technop\^ole de Ch\^ateau-Gombert} \\
{\sl 38 rue F. Joliot Curie, 13451 Marseille Cedex 20 - France} \\
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{\sl E-mail : angot@marius.univ-mrs.fr} \\
{\sl URL : http://www-irphe.univ-mrs.fr}
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\textbf{Abstract --}
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The concept of crack or contact resistance is sometimes introduced
empirically for diffusion pro\-blems with imperfect contact, e.g.
thermal or electrical contact resistance, or also hydraulic resistance
of fracture for flows in porous media. The objective is to take
account of fault lines or too thin layers compared to the largest
scale under study.  Hence, at this scale the solution is
\textit{discontinuous}.

Here, we generalize this concept and we formulate for such diffusion
problems a mathematical model with discontinuous coefficients which
includes an \textit{``imperfect contact'' transmission condition}
linking the diffusive flux with the jump of the solution.  It ensures
the well-posedness of the associated elliptic or parabolic problems
and we prove the global solvability within a variational framework.

For the numerical solution, we propose an original finite volume
method based on the introduction of \textit{``crack resistances''}
at the faces of the control volumes. In that way, we show how to satisfy
both conservativity and consistency of the numerical fluxes.
The convergence of this scheme is established within the
framework of finite volume theory. Some error estimates in ${\cal
  O}(h)$ are obtained for 1D or 2D rectangular meshes provided the
coefficients are piecewise of class ${\cal C}^{1}$ and the exact
solution is piecewise of class ${\cal C}^{2}$. Besides, the condition
number of the discrete operator is analysed and the improvement due to
the diagonal preconditioning is estimated.

Various numerical results illustrate the capabilities and the
efficiency of this method.


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