Summary: We apply the finite element method to the fourth non-mixed boundary value problem of the electrostatic field (in the sense of Galerkin). By changing the boundary value problem to an equivalent problem we give the conditions to guarantee the uniqueness of the weak solution (up to constants) to the fourth non-mixed boundary value problems. We also discuss the formation of the coefficient matrix and the right-hand vector of the corresponding algebraic equations and give an example to demonstrate our method.