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# 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

 (4.1)

The functions and are sufficiently smooth and of appropriate dimension. The upper subscript denotes the dependence on the stepsize. The optimization variable will comprise both the state and the control variables.

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 :
 (4.2)

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.

Subsections

Next: Neumann boundary conditions Up: paper84 Previous: Elliptic control problems with
Hans D. Mittelmann
2002-11-25