Resolution numerique de quelques problemes raides en mecanique des milieux faiblement compressibles

This paper deals with the numerical approximation of some «stiff» problems by asymptotic expansion of the solution. The model problem is the stationary linearized equation of slightly compressible fluids: $$\begin{gathered} \ll Find u_\varepsilon \varepsilon \left {H_0^1 (\Omega )} \right]^n s.t. \hfill \\ - \mu \Delta u_\varepsilon - \frac{1}{\varepsilon } grad div u_\varepsilon = f in \Omega \gg \hfill \\ \end{gathered} $$ where ∈ is asmall parameter; numerical treatment of the problem is thus difficult («stiff» problem). We establish existence and unicity of an asymptotic expansion foru, and use it to computeu. In the usual cases, with small divergence, the numerical results are far better than those obtained by direct discretisation of the problem. We also construct asymptotic expansions for the solutions of some nonlinear or non-stationary related problems.