Derivation of Finite Element Equations Using Variational Approach

This paper presents the derivation of the discrete finite element system for the Poisson equation. Based on the variational formulation, the domain is partitioned into sub-intervals to define piecewise linear basis functions. The coefficients of the stiffness matrix and the load vector are computed through integration over these intervals, leading to the construction of a linear system of equations. This process demonstrates how a continuous boundary value problem is transformed into a solvable matrix form.