  This paper is concerned with a rigorous convergence analysis of a fully
discrete Lagrangian scheme for the Hele-Shaw flow, which is the fourth order
thin-film equation with linear mobility in one space dimension. The
discretization is based on the equation's gradient flow structure in the
$L^2$-Wasserstein metric. Apart from its Lagrangian character --- which
guarantees positivity and mass conservation --- the main feature of our
discretization is that it dissipates both the Dirichlet energy and the
logarithmic entropy. The interplay between these two dissipations paves the way
to proving convergence of the discrete approximations to a weak solution in the
discrete-to-continuous limit. Thanks to the time-implicit character of the
scheme, no CFL-type condition is needed. Numerical experiments illustrate the
practicability of the scheme.
