A Study on orthogonal polynomials applied to the finite element method for solving differential equations

Abstract

The resolution of differential equations is fundamental in modeling phenomena in physics, engineering, and applied sciences. A classic example is the problem of the deflection of a rectangular plate, modeled by a boundary value problem. However, analytical solutions are limited to simple cases, which highlights the importance of numerical methods. Among the existing approaches, the finite element method stands out for its flexibility in representing solutions in complex domains. Its basic principle consists of obtaining an approximate solution within a projection subspace, onto which the differential equation is projected. Preliminarily, the mathematical foundations that support the numerical formulation were addressed, encompassing concepts from functional analysis, Hilbert spaces, and approximations of continuous functions by series of orthogonal functions. The central objective is to employ polynomials as basis functions in the finite element method, analyzing the convergence of the numerical solution through computational implementations carried out in GNU Octave. The results demonstrated stability and convergence of the solutions as the dimension of the projection subspace increased.