Discretization and optimization techniques

The discussion of discretization schemes is restricted to the standard situation where the elliptic operator is the Laplacian and the domain is the unit square The generalization to a general elliptic operator is straightforward. However, the modifications for an arbitrary domain depend essentially on the geometry of the boundary .

The purpose of this section is to develop discretization techniques by
which the problem (2.1)-(2.4) with
Neumann boundary conditions
resp. problem (3.1)-(3.4) with Dirichlet boundary conditions
is transformed into a nonlinear programming problem (NLP-problem) of the form

The form (4.1) will be achieved by solving
the elliptic equation (2.2) with the standard five-point-star
discretization scheme.
Choose a number
and the stepsize
Consider the mesh points

and consider the following sets of indices residing either in the domain or on the four edges of the boundary :

We have and . Denote approximations for the values of the state variables by for and denote approximations of the values of the control variables by for .

Now we shall specify the functions for the optimization problem (4.1) both for Neumann and Dirichlet boundary conditions.

- Neumann boundary conditions
- Dirichlet boundary conditions
- Optimization codes and modeling environment

2002-11-25