This chapter deals with the analysis of a non-conforming domain decomposition method, the mortar element method for solving systems of equations arising from Discrete Kirchhoff Triangles (D.K.T.) finite element discretization of a model fourth-order elliptic problem. We define the spaces of approximation and the decomposed discrete problem which of course relies on the variational formulation of the continuous one. We verify the uniform ellipticity and the uniform continuity (independent from the discretization parameter) of the discrete bilinear form. We present a mathematical analysis of the method and derive optimal consistency error and best fit error.