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.
