Fourth order elliptic singularly perturbed problems arise in linear elasticity of thin bucking plate, and in study the linearization of Cahn-Hilliard and Monge-Amperè equations. In this talk, we consider two kinds of boundary conditions. Firstly clarify the validity of its mixed finite element method, and present a new scale of the error between the exact solution and the numerical solution, whose each component is balanced with respect to the perturbation parameter. We then develop a novel analytical technique on  a posteriori error estimates based on an approximation result, and obtain robust residual-based a posteriori estimators in this scale. Numerical are presented to support the theoretical results.