With the help of a set of exact closed-form solutions to the stationary Gross-Pitaevskii equation, we compre-hensively investigate Landau and dynamical instabilities of a Bose-Einstein condensate in a periodic array of quantum wells. In the tight-binding limit, the analytical expressions for both Landau and dynamical instabilities are obtained in terms of the compressibility and effective mass of the BEC system. Then the stability phase diagrams are shown to be similar to the one in the case of the sinusoidal optical lattice.