The differential geometric algebra foundations developed in the previous paper \textit{Differential geometric algebra using Leibniz, Grassmann} can be used for creating a more universal language for finite element methods based on a discrete manifold bundle. Tools built on these foundations enable computations based on multi-linear algebra and spin groups using the geometric algebra known as Grassmann algebra or Clifford algebra. This foundation is built on a direct-sum parametric type system for tangent bundles, vector spaces, and also projective and differential geometry. Geometric algebra is a mathematical foundation for differential geometry, which can be used to simplify the Maxwell equations to a single wave equation due to the geometric product. Introduction of geometric algebra to engineering science disciplines will be easier with programmable foundations. In order to devise an expressive and performance oriented language for efficient discrete differential geometric algebra with the Grassmann elements, an efficient computer algebra representation was programmed. With this unifying mathematical foundation, it is possible to improve efficiency of multi-disciplinary research using geometric tensor calculus by relying on universal mathematical principles. Tools built on universal differential geometric algebra provide a natural geometric language for the Helmholtz decomposition and Hodge-DeRahm co/homology. Using the new \textit{Grassmann.jl} package, it is possible to compute anti-symmetric tensor products and geometric algebra in a high performance computing context. The abstract nature of the product algebra code generation enables extension of the product operations by way of Julia's type system, with which it is possible to construct customized mixed tensor products from a vector basis along with dyadics, as well as bivector elements of Lie groups. Given this mathemaical foundation, modular finite element methods based on geometric algebra with a discrete manifold can be built easily. Combined, these developed software tools provide a unified and universal foundation for discrete differential geometry, finite element methods, and quantum computing.