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In this article, we provide stability estimates for the finite elementdiscretization of a class of inverse parameter problems of the form$-\nabla\cdot(\mu S) = \g f$ in a domain $\Omega$ of $\R^d$. Here $\mu$ is theunknown parameter to recover, the matrix valued function $S$ and the vectorvalued distribution $\g f$ are known. As uniqueness is not guaranteed ingeneral for this problem, we prove a Lipschitz-type stability estimate in anhyperplane of $L^2(\Omega)$. This stability is obtained through an adaptationof the so-called discrete \emph{inf-sup} constant or LBB constant to a largeclass of first-order differential operators. We then provide a simple andoriginal discretization based on hexagonal finite element that satisfies thediscrete stability condition and shows corresponding numerical reconstructions.The obtained algebraic inversion method is efficient as it does not require anyiterative solving of the forward problem and is very general as it does notrequire any smoothness hypothesis for the data nor any additional informationat the boundary.

Stability for finite element discretization of some elliptic inverse parameter problems from internal data -- application to elastography