Numerical Solutions of Poisson Problem by Using Finite Element Method Based on Linear, Quadratic, and Cubic Polynomials

The finite element approach, which uses linear, quadratic, and cubic polynomials, can solve the elliptic form of the Poisson equation. This allows the method to provide a solution to the Poisson problem. This possibility arises from the structure within which the method operates. This work aims to provide an analysis of the technique in question and demonstrate its potential for applications in both square and circular dimensions. In conducting this investigation, a direct solution of the system of linear equations was used. Obtaining optimal error estimates in the L2 and H1 norms is possible for us if we choose to use numerical solutions and experimentation as our method of choice. This will enable us to achieve our goal. This article will present a range of examples that exemplify this concept, and its objective is to study the rates of convergence for cyclic and quadrilateral shapes. Additionally, the article will provide a variety of instances. The research presents a numerical representation of its findings. The research utilizes MATLAB R2018b to display the information in the form of graphics and tables. This has been done to provide a more realistic explanation of the facts presented.