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
Choose a number
and the stepsize
Consider the mesh points
Now we shall specify the functions for the optimization problem (4.1) both for Neumann and Dirichlet boundary conditions.