Browsing by Author "Angervuori, Ilari"

Purpose of the work is to give an elementary introduction to the Finite Element Method. We give an abstract mathematical formalization to the Finite Element Problem and work out how the method is a suitable method in approximating solutions of Partial Differential Equations. In Chapter 1 we give a concrete example code of Finite Element Method implementation with Matlab of a relatively simple problem. In Chapter 2 we give an abstract formulation to the problem. We introduce the necessary concepts in Functional Analysis. When Finite Element Method is interpreted in a suitable fashion, we can apply results of Functional Analysis in order to examine the properties of the solutions. We introduce the two equivalent formulations of weak formulation to differential equations: Galerkin’s formulation and minimizing problem. In addition we define necessary concepts regarding to certain function spaces. For example we define one crucial complete inner product space, namely Sobolev space. In Chapter 3 we define the building blocks of the element space: meshing and the elements. Elements consists of their geometric shape and of basis functions and functionals on the basis functions. We also introduce the concepts of interpolation and construct basis functions in one, two and three dimensions. In Chapter 4 we introduce implementation techniques in a rather broad sense. We introduce the crucial concepts of stiffness matrix and load vector. We introduce a procedure for implementing Poisson’s equation and Helmholt’z equation. We introduce one way of doing numerical integration by Gaussian quadrature points- and weights. We define the reference element and mathematical concepts relating to it. Reference element and Gaussian quadrature points are widely used techniques when implementing Finite Element Method with computer. In Chapter 5 we give a rigid analysis of convergence properties of Finite Element Method solution. We show that an arbitrary function in Sobolev space can be approximated arbitarily close by a certain polynomial, namely Sobolev polynomial. The accuracy of the approximation depends on the size of the mesh and degree of the polynomial. Polynomial approximation theory in Sobolev spaces have a connection to Finite Element Methods approximation properties through Cèa’s lemma. In Chapter 6 we give some examples of posteriori convergence properties. We compare Finite Element Method solution acquired with computer to the exact solution. Interesting convergence properties are found using linear- and cubic basis functions. Results seem to verify the properties acquired in Chapter 5.