We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 2017]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piece-wise smooth approximation of the matrices involved in a standard finite element discretization. In particular, it relies on the projection of smooth so-called stencil functions onto high-order polynomial subspaces. The performance advantage of the surrogate matrix methodology is seen in constructions where each stencil function uniquely determines the values of a significant collection of matrix entries. Such constructions are shown to be widely achievable through the use of locally-structured meshes. Therefore, this methodology can be applied to a wide variety of physically meaningful problems, including nonlinear problems and problems with curvilinear geometries. Rigorous a priori error analysis certifies the convergence of a novel surrogate method for the variable coefficient Poisson equation. The flexibility of the methodology is also demonstrated through the construction of novel methods for linear elasticity and nonlinear diffusion problems. In numerous numerical experiments, we demonstrate the efficacy of these new methods in a matrix-free environment with geometric multigrid solvers. In our experiments, up to a twenty-fold decrease in computation time is witnessed over the classical method with an otherwise identical implementation.