  In this article, we provide stability estimates for the finite element
discretization 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 the
unknown parameter to recover, the matrix valued function $S$ and the vector
valued distribution $\g f$ are known. As uniqueness is not guaranteed in
general for this problem, we prove a Lipschitz-type stability estimate in an
hyperplane of $L^2(\Omega)$. This stability is obtained through an adaptation
of the so-called discrete \emph{inf-sup} constant or LBB constant to a large
class of first-order differential operators. We then provide a simple and
original discretization based on hexagonal finite element that satisfies the
discrete stability condition and shows corresponding numerical reconstructions.
The obtained algebraic inversion method is efficient as it does not require any
iterative solving of the forward problem and is very general as it does not
require any smoothness hypothesis for the data nor any additional information
at the boundary.
