A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y ( iv) ( x) = f( x, y), a < x < b, y( a) = A 0, y″( a) = B 0, y( b) = A 1 y′( b) = B 1, the simple-simple beam problem. The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids. Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid. Modifications to these two formulas are obtained for problems with boundary conditions of the form y( a) = A 0, y′( a) = C 0, y( b) = A 1, y′( b) = C 1, the clamped-clamped beam problem. The general boundary value problem, for which the differential equation is y ( iv) ( x) = f( x, y, y′, y″, y‴), is also considered.