Mesh networks are among the most important network topologies in massive multiprocessor parallel systems. This paper proposes a new method for calculating the connectivity probability for mesh networks, which uses thorough mathematical methods to derive the relationship between the network node failure probability and the network connectivity probability in mesh networks. The results show that practical mesh networks can remain connected with very high probability. For example, it is formally proved that when the network node failure probability is bounded by 0.12%, the mesh networks with up to forty thousand nodes remain connected with probability larger than 99%. Our methods are also useful for deriving lower bounds on the connectivity probability for other computer networks.