  We study in this paper a multilayer discretization of second order elliptic
problems, aimed at providing reliable multilayer discretizations of shallow
fluid flow problems with diffusive effects. This discretization is based upon
the formulation by transposition of the equations. It is a Petrov-Galerkin
discretization in which the trial functions are piecewise constant per
horizontal layers, while the trial functions are continuous piecewise linear,
on a vertically shifted grid.
  We prove the well posedness and optimal error order estimates for this
discretization in natural norms, based upon specific inf-sup conditions.
  We present some numerical tests with parallel computing of the solution based
upon the multilayer structure of the discretization, for academic problems with
smooth solutions, with results in full agreement with the theory developed.
