Examining the best-fit paradigm for FEM at element level

Interestingly, an esoteric branch of pure mathematics called "functional analysis", more general and profound than variational calculus and originally developed by mathematicians, can be employed to explain clearly how the finite element machinery works. In a very abstract way, finite element results can be portrayed as "shadows" or orthogonal projections on predetermined function subspaces of the analytical results, known or unknown. Herein lies the philosophy of the finite element method. In the present work, an effort has been made to validate this important aspect of the finite element method at the element level through a particular differential equation representing a specific case of equilibrium.