Algorithmic approach to the finite element approximation method

The paper provides a general method for obtaining the so-called “shape functions”, which approximate the solution of systems of differential equations with boundary conditions. Unlike classic methods usually used for solving variational problems, the presented procedure approximates the unknown functions with piecewise linear functions, using a sequence of C++ code. The domain in which the system is defined is divided into finite elements, and the nodal values of the solution are obtained by solving the Fermat system associated with the specific boundary conditions of the defined domain. The proposed algorithm, based on C++ code, is designed to solve problems in two-dimensional cases or higher.