Finite elements introduction

Hi

Today I will give a small introduction about one of the most used numerical methods in science and engineering world: finite elements method.

Finite element method consist on purpose a discrete function as an approximated solution to an equation or set of equations describing a physic or mathematical system.

I wil ilustrate an unidimensional example very intuitive to see which is the basics:

Assume we watn to solve an equation like that

$$ \partial_x (p \partial_x y)+ (q+\lambda \sigma) \partial_x y= 0$$

This equation is a general expression for Sturm-Liouville problem, but this is irrelevant in this post. It can be observed that is a second order differential equation with variable coefficients.

Finite element method purpose the division of "y" function in n geometric elements. Extremes where two or more elements join are called nodes. We also assume that we know exactly the value of y funcion in each node. As we only know the value on the nodes, we need to interpolate the value of y in every part of the domain, that is, in each part of each element. For that purpose, we propose a "nodal function", which will take a 0 value for all points of the domain and 1 exactly in the node where is defined.

The next step is quite interesting. As we have divided the y function (the solution of the system) in n elements, we have n+1 nodes that they are connected because they are defined in our domain of study. We can assume then that the global solution will be a linear combination of each nodal function defined for each element that forms our domain. Constants of linear combination will be the n+1 values of the funcion exactly known in each node.

$$y=\Sigma_{n=0}^{n+1} \phi_n y_n$$

Notice that the approximated discrete solution, if solution, must satisfy the differential equation. So:

$$ \partial_x (p \partial_x (\Sigma_{n=0}^{n+1} \phi_n y_n))+ (q+\lambda \sigma) \partial_x (\Sigma_{n=0}^{n+1} \phi_n y_n)= E $$

It can be appreciated that the equation is not equal to zero: this is true, as we have approached a continuous  solution with a discrete funcion. There must be an error E by doing this. And it can be seen that we have just obtained the expression of the error of our method. The next step consist of choosing a good criteria to make converge this error funcion  E to zero. If we success in our task, we will have a good solution for our equation.

The method we are going to use is called "average weight method. This method try to find a serie of functions such us its scalar product with error function must be zero.

$$\int w(x) E dx = 0$$

We just want to find a set of functions orthogonal  to error function. If we chose weight functions as the same type as nodal functions, we can just see that by doing integrals converge to zero, we are also doing our error funcion converge to zero.

This method is quite simple in its basic structure, but its computing is quite hard to compile. We know that the more nodes we consider, the more exact our solution will be, but we have to compute more calculations. The problem is not so trivial as we are limited by our computation capacity.


In following posts I will explain in deep some examples and other detailed observations about this method.