Discrete geometric analysis is a discrete version of geometric analysis. When we develop discrete geometric notions, it is important to compare them with corresponding geometric notions defined on a Riemannian manifold. In the case of a crystal lattice, a periodic lattice in the d-dimensional Euclidean space, we change scales smaller and smaller, it converges to a d-dimensional vector space, and random walks on a crystal lattice converges to the Brownian motion of the vector space with the Euclidean metric induced by the standard realization of the crystal lattice. In the case of a discrete surface, in the 3-vector space, we cannot apply that scheme because it does not have a linear structure. Instead of scale change, we subdivide the discrete surface to have a finer discrete surface. In that way, we have a sequence of subdivided discrete surfaces, and discuss convergence of geometric notions. We share examples of applications of the convergence theories to the study of materials research.