Nonlinear finite elements/Solution of Poisson equation

Construction of Approximate SolutionsEdit

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 equationEdit

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.