A Finite Element Method Framework for Solving Schr¨odinger Eigenvalue Problem in 2D رسالة ماجستير

Abstract

The Finite Element Method (FEM) is considered as well-known technique for solve partial differential equations numerically. It is also originally developed for solving problems occur in engineering and mathematical physics. Moreover, it is a numerical method for solving complex problems like structural analysis, heat transfer, fluid flow and complex systems. In Finite Element Method, we divide the domain in one direction into cells and use polynomials for approximating a function over a cell. Whereas, if the domain has two directions we divide it into intervals sub-domains and a given system discretization into smaller, simpler parts that are called elements and these are connected by Nodes and we choose the basis functions to provide an approximations of the unknown solution within an element. The Finite Element Method formulation of a boundary value problem finally leads in a system of algebraic equations. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations. The basis functions has three different kinds of elements which are linear , quadratic element and cubic element. In this work, we use only a linear elements because it a quicker than others. We apply the finite element method with double well-potential using three different domains. at The end of the thesis, we give a numerical examples with different domains.