Field electron emission systems are described by the Poissons equation for electrostatic potential. For the simple vacuum microdiode, consisting of the cathode and the anode, it is supposed that input data of equation have inherent errors of measurements. Numerical solution of a two-dimensional Poisson equation together with Dirichlet boundary conditions is reduced to the solution of an interval system of linear algebraic equations. The algorithm of two-sided estimation of an error of solution is formalized on two point-wise matrices of the system by the analysis of scalar component-wise product of elements of a matrix and their cofactors.