Finite Element Method for Elliptic Differential Equations

Abstract

Abstract In this thesis we will discuss some basic and general theory of the finite element method. We will also discuss the variational formulation and discretization in order to assess the amount of error in the approximate solution applied to the space segmentation into triangles. For this purpose, first we are going to study the finite element method for second order elliptic problems in one and two dimensions and find a posteriori error estimates for Reaction-diffusion problems and Poisson equation. After that, we will review the modular solution method and the system of fragmentation of differential equations in different conditions on the limits of the definition range. Also illustrative examples will also be analyzed using the mathematical programming language ’Matlab’. The a posteriori errors reviewed in this thesis are quantities that measure the rate of convergence of the numerical solutions of differential equations to the exact solution using a particular element method that can be estimated based on the approximate solution and the information available on differential equations. The advantage of the numerical errors of differential equations is to measure the size of the error in order to make it as small as possible and thus get the best approximation of the solution. To discuss these errors, there are basic concepts that will be addressed to explain, and then the errors will be reviewed in details for some partial differential equations.