the finite element method is a powerful tool for approximating the solutions of a large class of pde's. traditional fem requires the function spaces to be linear. we discuss an extension of fem, geodesic finite elements, which allows the functions being approximated to have range in a non-linear riemannian manifold. we realize the set of pseudo riemannian metrics as a symmetric space in order to pull back the matrix exponential onto it, thus endowing it with a riemannian metric. once done, we are able to discretize a class of pseudo-riemannian manifolds. this has possible applications in non-linear hyperbolic pde's.