Followed by this topic where I discussed how to use Finite Difference Method in numerical analysis. Today, we will study a so-called Finite Element Method, an improved Finite Difference Method. We first consider the following problem
where with boundary conditions .
The above problem is usually called 2-point boundary value problem of laplacian equation in 1D.
We first consider such problem by using Finite Difference Method. The idea of using the Finite Difference Method is to split equally the interval into pieces by where . We then approximate by using the following
where . Therefore, we have the following system of linear equations
The idea of the Finite Element Method is to split the interval into pieces by where but not necessarily equality. In order to deal with the problem, we shall look for weak solutions. In the sense of the distribution, is called a weak solution if
Under some conditions, for example, if , the existence of weak solution is well-understood via the Lax-Milgram lemma.
Let be a collection of piecewise linear map in so that is linear on every intervals where . We also suppose that is such that . Clearly the space is finite dimension with a basis constructed as follows
It is easy to see that .
We will find as a linear combination of these functions denoted by , to be exact, we will approximate by such that at for every . Precisely, let
we then have
From the equation defining the notion of weak solution we replace by and by , we now get
The above identity can be rewritten as a linear system as the following
where we use the following notation