Algebraic-topological chains defined over finite cell complexes have been proposed as a uniform computational means of representing physical objects, systems and properties. In this article, we introduce CHAINS, an algebraic-topological computer language for representing and computing the properties of a wide variety physical systems. In particular, we develop a CHAINS program that implements a finite element approximation to linear elasticity. In the process we illustrate the relationship between finite element analysis and the chain models methodology, showing that while finite element analysis is a specific numerical approximation scheme, CHAINS is a computer language for specifying and representing a variety of physical systems and approximations, of which finite element computations are one example.