# Finite elements/Solution of Poisson equation

## Construction of Approximate Solutions

[edit | edit source]If we know that the problem is well-posed but does not have a closed form solution, we can go ahead and try to get an approximate solution. The finite element method is one way of getting at approximate solutions (among many other numerical methods).

The finite element method starts off with the variational form (or the
weak form) of the BVP. The method is a special case of a class of
methods called ** Galerkin methods**.

## Finite element solution for the Poisson equation

[edit | edit source]Recall the variational boundary value problem for the Poisson equation:

```
```

The space is continuous and an infinite number of functions could
be chosen from this space of functions. In the finite element
method, we choose a trial function from the space of approximate
solutions where . A defining feature
of these approximate trial solutions is that they are associated with
a ** mesh** or ** discretization** of the domain . These
functions also have the feature that they are finite dimensional with
each dimension being associated with a ** node** on the mesh.

Assume that we are given . Let us choose a weighting function that satisfies on . We can choose another function as our trial solution. Since the boundary condition on is , both and can have the same form. In the next section, we will look at the general form of the heat equation where on the boundary.

In finite element methods we choose trial solutions of the form

where , , , are nodal temperatures which are
constant on . The functions form a
basis that spans the subspace and are known as
** basis functions** or ** shape functions**.
Note that is the total number of nodes minus the number of nodes
on where is specified.

Since the functions come from the same space of functions, we can represent them as

where , , , are arbitrary constant on with the restriction that on .

If we plug in these finite dimensional forms of and into the variational BVP, we get an approximate form of the variational BVP which can be stated as:

```
```

After substituting the expressions for and in the variational BVP we get

where,

In matrix form, we have

where , is a symmetric matrix, is a vector, and is a vector.

Since can be arbitrary, equation (38) can be further simplified to the form

This system of equations has a solution since is positive-definite and therefore has an inverse. Once the s are known, the approximate solution can be found using

The functions have special forms in the finite element method that have the property that the quality of the approximation improves with an increase in the dimension of the basis.