This paper investigates the analytical solutions and numerical solutions of second order linear boundary value problems with three-point boundary conditions by means of the weak pre-orthogonal adaptive Fourier decomposition (W-POAFD) method introduced by Qian et al. Using the reproducing kernel functions of the Sobolev spaces on the unit interval (Formula presented.), we construct a dictionary to implement the W-POAFD. Based on orthonormalizataion, we can select the crucial parameters successively by adopting the weak maximal selection strategy of the W-POAFD method. Then the numerical solutions have “sparse” representations, which converge to the analytic solution with fast convergence rate. Some numerical experiments demonstrate the high accuracy and efficiency of the W-POAFD method in this three-point boundary value problem.