In this paper we consider a numerical solution of two boundary value problems: (BVPl) $-u''(x)=f(x,u)$, $u(0)=y_0$, $u(1)=y_1$ and (BVP2) $-u''(x)-q(x)u(x)=f(x)$, $u(0)=y_0$, $u(1)=y_1$, using the authors' four-point rule of degree 3 for the second derivative. The discretisation meshes depend upon this rule. The discrete problem to (BVP1) has the usual form, but the discrete problem to (BVP2) has an unusual form. Under certain assumptions, our schemes have a third order of convergence.